Could anyone please explain the difference between the conduction current density $J=σE$ and the convection current density $J=ρv$? I really appreciate any examples or applications to further elaborate these two ideas.
Note: $v$ is the particles' average drift velocity.
The equation $$\vec J = \rho \vec v$$ where $\vec J$ is the current density (charge / time / area), $\rho$ is the charge density ( charge / volume ) and $\vec v$ is the velocity (distance / time) at a point in a fluid. It simply relates two quantities (a velocity and a charge density) into a flux of charge across a surface. It is a definition, with the only real assumption having to be non-relativistic speeds and inertial frames of reference.
Conversely, Ohm's Law $$ \vec J = \sigma \vec E $$ where $\sigma$ is the conductivity of the fluid and $\vec E$ is the electric field. It's name is a bit of a misnomer as it is not a law in the strict sense; it has a narrow range of applications involving small velocities, low-mass charges (i.e. electrons vs protons), and low magnetic fields. It also assumes the conductivity $\sigma$ is the same in every direction (isotropic). So you see, lots of assumptions.
For example consider a simple plasma of electrons and protons with weak magnetic fields and not too hot. Both the electrons and protons are moving charged particles, so $\vec J_e = -\rho \vec v_e$ and $\vec J_p = \rho \vec v_p$ both apply (with a minus sign for the electrons negative charge). But the electrons move much faster than the protons, have smaller masses, and don't exert much pressure, so Ohm's law $\vec J_e = \sigma \vec E$ could apply to them (in other words their motion is completely dominated by the electric field and bouncing off of the protons). The motion of the protons however may be more dominated by the pressure in the plasma and so follow more closely the behavior of a fluid, and Ohm's law is not appropriate to describe that behavior.
In either case you are looking at current density. When you express it in terms of conductivity and electric field, you express the fact that the charge carriers are moving under the influence of an electric field.
If you have charges moving under the influence of "any" force, and you know their velocity, you can compute the current density if you know the density. That leads to the second equation.
Combining them, if you know the density, conductivity and electric field, you can compute the velocity. Same thing is true for any other three known parameters allowing you to compute the fourth.
