While studying about the magnetic properties of matter, my book defines this vector as
$\vec{H}=\dfrac{\vec{B}}{\mu_0}-\vec{M}$
where $B$ is the magnetic field and $M$ is the intensity of magnetisation.
However it did not provide any definition of this new physical quantity whatsoever.
$\vec{H}$ is sometimes called the magnetic field strength (I prefer "H-field"). More commonly $H$ will confusingly be called the magnetic field -- we will return to this.
As you likely know $\vec{B}$ is just the typical magnetic field.
The key of this is $\vec{M}$. $\vec{M}$ is the magnetization, which represents the amount of magnetic dipoles in material. These dipoles can be either permanent (i.e., ferromagnetic dipoles, resulting from spin) or induced dipoles due to an external magnetic field. You can think of $\vec{M}$ as the magnetiostatics analogue to the polarization in electrostatics, if you are familiar with that.
With $\vec{B}$ and $\vec{M}$ we can compute $\vec{H}$ using the formula in your question. Now note that though $H$ is called the magnetic field strength, it actually has units of linear current density (amperes/meter). $H$ is often called the "magnetic field strength" or irritatingly, the "magnetic field", because it takes the total magnetic field and subtracts off the contribution from a magnetized material. Thus, $\vec{H}$ quantifies the magnetic field strictly from free currents only.
$H$ is analogous to the electric displacement $\vec{D}$.
