Definition of current density

The equation is correct except that usually it is written $vec J=rho vec v$ where $rho$ is the charge density instead of the number density. So I will use the standard expression.

The SI units of $rho$ are $text{C}/text{m}^3$ which looks like it will give a volume density. But the units of $vec v$ are $text{m}/text{s}$ so the overall units of $vec J$ are $text{C}/text{m}^2text{s}$. Thus, the formula indeed gives an area density, meaning the current density integrated over an area gives the current flowing across that area.

The current density is defined by the current per unit area. If you multiply by the area (being crossed) you get the total current. In this case if you multiply the charge density by velocity (distance / time), you get the charge which will cross the unit of area in a unit of time.