From Wikipedia:
"The term (Magnetic Field) is used for two distinct but closely related fields denoted by the symbols $B$ and $H$, where $H$ is measured in units of amperes per meter in the SI. $B$ is measured in teslas in the SI."
So, the two are closely related. Why do we need two then? Could just one be used?
As I remember from university, for vacuum, Maxwell's equations are usually written in terms of $B$, while for media in terms of $H$ (and $B=\mu H$).
In layman's terms,
E and B are the total electric and magnetic fields.
D and H are the free electric and magnetic fields.
P and M are the bound electric and magnetic fields.
M would be the magnetic field caused by current loops in the material. In vacuum, like you said, B and H are proportional by a constant since there is no material. However, when you are not in a vacuum, you would need to incorporate M, leading to the equation B = H + M in natural units.
I love this question! Because I've struggled with it before, coming out frustrated that no one gave me the easy explanation. :-)
Now, I'm not a physicist, but I think I've managed to learn the correct intuition here:
$\vec{D}$ and $\vec{B}$ are electric & magnetic flux densities.
$\vec{E}$ and $\vec{H}$ are electric & magnetic field strengths.
The difference? Flux doesn't depend on the material, but field strength does — recall Gauss's law: $$Q = \oint_S \vec{D}\cdot \,d\vec{A}$$
Flux only depends on the charge inside your closed surface. (The "flow" must leave the volume!)
But naturally if you change the material then something is affected — and that's the field strength.
If you ever forget, just remember the units:
$\vec{D}$ is in $\text{C}/\text{m}^2$, hence there's no $\epsilon$.
$\vec{B}$ is in $\text{Wb}/\text{m}^2$, hence there's no $\mu$. (Though honestly I remember this by analogy with $\vec{D}$.)
The fields $\bf E$ and $\bf B$ are the fundamental components of the electromagnetic field. They define the field via its effects, producing a force on a charge $q$: $$ {\bf f} = q ( {\bf E} + {\bf v} \times {\bf B}). $$ One can in principle do the whole of electromagnetism using just these fields.
In a material medium (e.g. glass, or a metal, or a semiconductor, or a gas, etc.) these fields are very complicated however. They vary at lot on the distance scale of the atomic spacing, getting huge near atomic nuclei and smaller elsewhere. So most of the time we do not treat the fields at a point, but rather we average over a region of space of the order of a few atomic spacings (e.g. a nanometre in a solid, or a larger region in a gas). This is where other things such as polarization $\bf P$ and magnetisation $\bf M$ come into play. If we divide the charge in any region into the part associated with little electric dipoles (called bound charge) and the rest (called free charge) then we have $$ q_{\rm tot} = q_{\rm bound} + q_{\rm free} $$ so the first Maxwell equation reads $$ {\bf \nabla} \cdot {\bf E} = \frac{q_{\rm b} + q_{\rm f}}{\epsilon_0} $$ where I used the subscripts $b$ and $f$ for 'bound' and 'free'. Now a basic result (which can be proved with a little standard manipulation, it is first year undergraduate level) is that $$ {\bf \nabla} \cdot {\bf P} = -q_{\rm b} $$ where $\bf P$ is the dipole moment per unit volume, called polarization. So it follows that $$ {\bf \nabla} \cdot (\epsilon_0 {\bf E} + {\bf P}) = q_{\rm f}. $$ Well look at that! The right hand side is nice and simple, because it only involves the free charge, and it often happens that we know from the start how much free charge there is. There might even be none at all! (e.g. a light wave travelling in glass in ordinary circumstances). So we choose to give the combination $(\epsilon_0 {\bf E} + {\bf P})$ a symbol of its own: we call it $\bf D$. This is how the field $\bf D$ gets introduced, with its associated equation $$ {\bf \nabla} \cdot {\bf D} = q_{\rm f}. $$
The story for $\bf H$ is similar. First we derive by analysis that the magnetization is connected to the part of the total current that is caused by little current loops, with other contributions coming from free current and changes in the dipoles. The central result of this derivation is that the total current can be divided up as $$ {\displaystyle \mathbf {j}_{\rm tot} =\mathbf {j} _{\mathrm {f} }+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}}. $$ Next we use this in the fourth Maxwell equation, and we notice that a convenient way to write the equation is to give the combination ${\bf B}/\mu_0 - {\bf M}$ its own letter ($\bf H$) and we arrive at $$ {\bf \nabla} \times {\bf H} = {\bf j}_{\rm f} + \frac{\partial \bf D}{\partial t}. $$ Again, this is convenient because often we want to treat problems without worrying about what the magnetisation current and the bound charge is doing.
So, to conclude, fields $\bf E$ and $\bf B$ are the basic fields. (Together they make up a tensor called the field tensor, but you don't need to know that.) Fields $\bf D$ and $\bf H$ are introduced for reasons of mathematical convenience and the associated physical insight. They are particularly useful when thinking about capacitors and inductors and electromagnetic waves propagating in media when the free charge and free current are known.
The other two Maxwell equations do not involve the sources so they are not affected. They are $$ {\bf \nabla} \cdot {\bf B} = 0 $$ and $$ {\bf \nabla} \times {\bf E} = - \frac{\partial \bf B}{\partial t} $$ Note, for example, it is $\bf B$ not $\bf H$ which has zero divergence. So $\bf B$ lines always run in closed loops, but $\bf H$ lines do not need to if there is some magnetization around. In a similar way, if some medium carries no free charge then the $\bf D$ field runs in closed loops (or it may be zero), but the $\bf E$ field might not.
On usefulness
The fields $\bf D$ and $\bf H$ are useful when considering things like capacitors and inductors, but they really come into their own when considering electromagnetic waves in dielectric media. It would be hard work calculating things like reflection coefficients without them. And they also come into their own in the consideration of energy. The energy flow for example is given by the Poynting vector $$ {\bf S} = \frac{1}{\mu_0} {\bf E} \times {\bf B} - {\bf E} \times {\bf M} $$ ---a rather tricky formula. But how much easier it is in terms of $\bf E$ and $\bf H$: $$ {\bf S} = {\bf E} \times {\bf H}. $$
On relative permittivity and permeability
We always have ${\bf \nabla} \cdot {\bf H} = - {\nabla} \cdot {\bf M}$ and ${\bf \nabla} \cdot {\bf B} = 0$. But this means it will often not be possible to write ${\bf B} = \mu_0 \mu_r {\bf H}$, so answers which only make reference to that formula are missing a major part of the physics. In particular you usually can't use ${\bf B} = \mu_0 \mu_r {\bf H}$ when thinking about permanent magnets.
In the case of a permanent magnet you have a static case with no free current, so ${\bf \nabla} \times {\bf H} = 0$, which means the integral of ${\bf H}$ around a loop is zero, but this will not be true for $\bf B$, so there is no simple proportionality between them. However, in many simple amorphous media it happens that, at low fields, ${\bf M} \propto {\bf H}$. So in this case ${\bf B}$ is also proportional to ${\bf H}$ so we can introduce the relative permeability $\mu_r$ defined through the equation $$ {\bf B} = \mu_0 \mu_r {\bf H}. $$ This is a useful equation, but much more restricted in its validity than the other ones I have written above. (Actually we can also use this result slightly more generally, in non-linear materials where $\bf M$ is not proportional to $\bf H$ but is in the same direction; in this case $\mu_r$ will depend on ${\bf H}$.) Similarly, simple dielectric media will have polarization proportional to $\bf E$, and consequently $\bf D$ proportional to $\bf E$, so we define a relative permittivity $\epsilon_r$: $$ {\bf D} = \epsilon_0 \epsilon_r {\bf E}. $$
But what is the difference between $\bf B$ and $\bf H$ in physical terms?
${\bf B}$ is the field which gives the force on a moving charge, and it is the field which is induced by a changing electric field. It is the one involved in electromagnetic induction. Its integral over a surface is the flux.
$\bf H$ is the field which is easily calculated from a given amount of free current, and the component of $\bf H$ along a boundary does not change when you move from one medium to another (if there is no surface free current). This makes $\bf H$ useful in calculating what electromagnetic waves do, and it is also useful for tracking energy movements via the Poynting vector ${\bf S} = {\bf E} \times {\bf H}$.
A permanent magnet has $\bf B$ running in loops and $\bf H$ following $\bf B$ outside, but not inside, the magnet, in such a way that its integral around a loop is zero (unless there is a current flowing nearby), c.f. Direction of H and B inside and outside a bar magnet
A lump of glass with light waves propagating in it has both $\bf B$ and $\bf H$. If you have an inductor made of a solenoid with a fixed current, then when you slide a piece of glass into the cylinder (keeping the current constant) the value of $\bf H$ does not change but the value of $\bf B$ does. And if you slide in a piece of soft iron the value of $\bf B$ changes enormously. In this case the current supply providing the constant current will do some work, which provides the field energy.
A comment on units and physical dimensions
One of the puzzles of this area of physics is why $\bf B$ and $\bf H$ have different physical dimensions in the SI system of units, and so do $\bf D$ and $\bf E$. One should not hang too much on that. It is just a human choice about definitions. The people inventing the SI system could, with good logic and physical sense, have chosen to introduce the field ${\bf B} - \mu_0 {\bf M}$, giving it the symbol say $\tilde{\bf H}$. Then we would all be learning the formulae $$ \tilde{\bf H} = {\bf B} - \mu_0 {\bf M} $$ and $$ {\bf \nabla} \times \tilde{\bf H} = \mu_0 {\bf j}_{\rm f} + \mu_0 \frac{\partial {\bf D}}{\partial t}. $$ This way of looking at things has the handy result that $\tilde{\bf H}$ and $\bf B$ have the same physical dimensions, which makes a lot of sense, but it results in a $\mu_0$ in the formula relating $\tilde{\bf H}$ to current and the inventors of the system of units wanted to avoid that. So there we have it.
Here is a reason.
The fourth of Maxwell's macroscopic equations says that $$ \nabla \times \vec{H} = \vec{J} +\frac{\partial \vec{D}}{\partial t},$$ where $\vec{J}$ is the free current at a point. In general, it is not possible to rewrite this in terms of B-field without a detailed knowledge of the microscopic behaviour of the medium (with the exception of vacuum) and what currents and polarisation charges are present, either inherently, or induced by applied fields. Sometimes the approximation is made that $\vec{B} = \mu \vec{H}$, but this runs into trouble in even quite ordinary magnetic materials that have a permanent magnetisation or suffer from hysteresis and the general relationship is that $$ \vec{B} = \mu_0 (\vec{H} + \vec{M}) , $$ where $\vec{M}$ is the magnetisation field (permanent or induced magnetic dipole moment per unit volume). For these reasons, the auxiliary magnetic field strength $\vec{H}$ is invaluable for performing accurate calculations of the fields induced by currents, or vice-versa, within magnetic materials.
On the other hand, the Lorentz force on charged particles is expressed in terms of the magnetic flux density $\vec{B}$. $$ \vec{F} = q\vec{E} + q\vec{v}\times \vec{B}$$ Indeed this can form the basis of the definition of B-field and can be used, along with the lack of magnetic monopoles, to derive Maxwell's third equation (Faraday's law), which does not feature the H-field. So, both fields are a necessary part of the physicists toolbox.
As Philosophiae Naturalis points out in a comment, the B-field can be thought of as the sum of the contributions from the (applied) H-field and whatever magnetisation (induced or intrinsic) is present. Often, we can only control or easily measure the applied H-field. In limited circumstances we can get away with using only one of the B- or H-field if the magnetisation is related to the applied H-field in a straightforward way. For other cases (and hence most ferromagnetic materials or permanent magnets) both fields must be considered.
Write down the Ampere's Law in vacuum:
$$\nabla \times \bf{B} = {\mu _0}\left(J + {\varepsilon _0}{{\partial E} \over {\partial {\rm{t}}}}\right)$$
Divide both parts by ${\mu _0}$ and substitute $D$ for ${\varepsilon _0}E$ to get:
$$\nabla \times \bf{B \over {{\mu _0}}} = J + {{\partial D} \over {\partial {\rm{t}}}}$$
So, I guess, it was very convenient to "get rid" of ${\mu _0}$ by defining $\bf{H}=\bf{B \over {\mu _0}}$ to get the Ampere's Law for a medium:
$$\nabla \times \bf{H} = J + {{\partial D} \over {\partial {\rm{t}}}}$$
No way I claim that this is how H (or B) appeared historically, but it is a way for me to remember the difference at least.
UPDATE: I received a downvote likely for stating that $\bf{H}=\bf{B \over {\mu _0}}$. So, disclaimer: this is, in general, not true. It was stated for vacuum.
$\mathbf{B}$ is the magnetic field in a vacuum. It is what would be most properly referred to as "magnetic field", by analogy with the "electric field". It's what, along with the electric field, directly governs the motion of charges.
$\mathbf{H}$ is an effective magnetic field that comes about when we consider a magnetic field (i.e. $\mathbf{B}$) penetrating a material object, but at a macroscopic scale where we can ignore the fact that matter is composed of tiny particles moving about with vacuum between them. Because the particles are electromagnetically active. In effect, it's what the interactions between $\mathbf{B}$, all the atoms and molecules in the material, and subject charge, "looks like" to said charge when moving through it at a scale which is far larger than those constituents. It lets you calculate the magnetic effect on such without having to worry about the details thereof.
Likewise, the same holds for the electric fields $\mathbf{E}$ and $\mathbf{D}$.
$\mathbf{B}$ and $\mathbf{E}$ are the more fundamental entities, while $\mathbf{H}$ and $\mathbf{D}$ are derived, or emergent, entities.
The field $\mathbf{B}$ is the one that matters. In vacuum, both $\mathbf{B}$ and $\mathbf{H}$ are same except, of course, for the constant permeability. One can say that $\mathbf{H}$ was invented to make things simple that is with free currents one can calculate $\mathbf{H}$. $\mathbf{B}$ is important when one considers fields in matter. That is where one has magnetic moments from matter. It would be wrong to consider $\mathbf{B}$ and $\mathbf{H}$ as separate entities. Note that whereas the field lines of $\mathbf{B}$ are closed, those of $\mathbf{H}$ are not necessarily closed.
It turns out that ampere law is still true in case of magnetics placed in external fields but we would have to include current due to magnetization.
$$ \oint B \cdot dl = \mu_o \left[ I + I' \right] \tag{1}$$
But now, we run into a problem, the magnetization current $I'$ is experimentally difficult to determine, how we can write equation of another field who's line integral over a loop is only determine by the current through the material. Applying the result of $ \oint J dl = I'$ on equation(1) and rearranging:
$$ \oint ( \frac{B}{\mu_o} -J) \cdot dl=I$$
Now, the quantity in integrand can be taken as:
$$ H = \frac{B}{\mu_o} - J$$
Now, this new vector field is independent of magnetization effects which makes it 'nice'.
Example of applying the result(page-181):
Consider a system of a long shape wire (the curve $\Gamma$ ) and an arbitary piece of paramagnetic through which the wire passes. Now, consider the line integral of magnetic field of over the loop $\Gamma$ notice that it is depending on if we place the paramagnetic or not. However, for the field $H$, the loop integral is same in both cases because it only depends on the conduction currents.
*:Based on discussion from page-178 to page-179 in I.E Irodov's Basic laws of electromagnetism.
$B=\mu H$
Where $\mu$ is the magnetic permeability of the material.
That's it. There's a lot more handwavium and complicated terminology, but that generally doesn't add anything of value.
(For a list of exceptions to this, look at the people screaming in the comments below.)
