Current Density comes in three forms:
1. Line Current Density, represented as Jl, is simply current, I, and is measured in Amperes (A). This is the amount of charge, q, passing through a point in an amount of time:

I = dq/dt

Jl = I

It is worth noting that Jl is a vector quantity, but we usually think of current as a scalar. This is because we usually indicate the direction of current flow as a simple arrow on a circuit diagram--so current really does have a direction associated with it, whether it is clockwise/counterclockwise or "in the +ax direction".

2. Surface Current Density, Js, measured in Amperes per meter (A/m). This is the current flowing along the surface of an object, say the outer surface of a wire. We now measure this by taking the amount of charge q, passing through a plane bisecting the surface in an amount of time:

Js = I/L

Where I is the total current, and L is, say, the circumference of the wire (or some other linear measurement of the suface through which the current is traveling).

Note that a surface charge density only occurs in approximations: if the surface through which current is flowing is infinitesimally thin, or if the material is a perfect conductor. Otherwise, current will leak into the volume of the medium through which it is traveling, which brings us to:

3. Volume Current Density, Jv, measured in Amperes per meter squared (A/m2). Here the charge is traveling throughout the entirety of the media, including the cross-section. We measure this by taking the charge passing through a plane bisecting the volume in an amount of time:

Jv = I/A

Where A is the cross sectional area of the wire (for instance). (But this is only true for uniform current distributions).

This is the most real of the Current densities, and is usually what is meant. Note that the current density in this case can often vary with such things as radius of the wire: it could be that (for instance) the current is more dense towards the outside of the wire, and less dense within.