Huh? Again, let's set things straight. First of all about dimensions. It depends on the system of units used, which dimensions quantities take. Let's use the SI units (which I don't like to much for doing theory, but which are most simple for the purpose here). In the SI on the level of electromagnetism one uses four base units: metre for length, seconds for time, kilo grams for mass, and Ampere for the electric current. More logical is to start from charges than from currents, i.e., here I'll use 1C=1 A s (Coulomb) as the unit of charge.
Now the dimensions are easily determined: One starts with charge density. That's a simplification in the sense that you don't take into account that in fact matter has to be described as consisting of particles, described by quantum field theory, but you take a classical point of view (which can to a certain extent be derived from the underlying fundamental microscopic physics) of continuum mechanics, i.e., you consider matter carrying electric charge as a continuum, i.e., you define charge density as a field ##\rho(t,\vec{x})## which tells you that at time ##t## a volume element ##\mathrm{d}^3 \vec{x}## at the place ##\vec{x}## contains a charge of $$\mathrm{d} Q=\mathrm{d}^3 \vec{x} \rho(t,\vec{x}).$$
Now the dimensions are clear: The volume element has dimension ##\mathrm{m}^3## (cubic metres) and the charge has dimension ##\mathrm{C}## (Coulombs). Thus charge density has the dimension ##\mathrm{C}/\mathrm{m}^3##.
Now you want to describe charge conservation in a local way, i.e., you want to describe that when the medium, carrying the charge, is moving the total amount of charge doesn't change. Now look at a charge ##\mathrm{d} Q=\mathrm{d}^3 \vec{x} \rho(t,\vec{x})## and think how it changes with time (keeping the volume element fixed). On the one hand the change is simply given by
$$\frac{\mathrm{d} Q}{\mathrm{d} t}=\int_{V} \mathrm{d}^3 x \partial_t \rho(t,\vec{x}).$$
On the other hand, charge conservation tells you that the only way the charge within the volume element can change is that charge flows in or out of this volume element through its boundary ##\partial V##.
To this purpose you define surface-element normal vectors along this surface, ##\mathrm{d}^2 \vec{f}##. Think of it to take a tiny ("infinitesimal") part of the surface and put a vector with the length given by the area of this surface element and in the direction perpendicular to the surface element. By definition these normal vectors are always pointing out of the volume. Now it's clear that the amount of charge is flowing out of the surface element within a time interval ##\mathrm{d} t## which is in the volume ##\mathrm{d} t \vec{v} \cdot \mathrm{d}^2 \vec{f}##. So charge conservation means
$$\frac{\mathrm{d}}{\mathrm{d} t} Q= \int_V \mathrm{d}^3 \vec{x} \partial_t \rho(t,\vec{x})=-\int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{v}(t,\vec{x}) \rho(t,\vec{x}).$$
Using Gauss's theorem and since you can do this for any volume, you get the local current-conservation law
$$\partial_t \rho=-\vec{\nabla} \cdot \vec{j},\quad \vec{j}=\rho \vec{v},$$
and from the meaning of ##\vec{j}=\rho \vec{v}## you see that this is a current-density vector with the dimension ##\mathrm{C}/(\mathrm{s} \; \mathrm{m}^2)##, i.e., for an arbitrary surface element ##\mathrm{d}^2 \vec{f}## you get the amount of charge flowing through this surface element per unit of time by ##\mathrm{d}^2 \vec{f} \cdot \vec{j}##. If it's positive, it means the flow is in direction of the surface-normal vector, otherwise it's against it.
Now sometimes you have other types of charge distributions, which are again simplifying idealizations. E.g., if you have a conductor, on which you put some charge, in the static case these charges will not fill the volume of this conductor but try to get as far from each other as possible, i.e., it will distribute mainly along the surface of the conductor. Idealized you can say it's just sitting on the surface. Then it makes sense to define the surface-charge density ##\sigma(t,\vec{x})## along the surface, i.e., it gives the charge per area at the point ##\vec{x}## of the surface. The dimension is ##\mathrm{C}/\mathrm{m}^2##.
The same holds if your conductor is a very thin wire. Then you can idealize the charge distribution to be just along this wire, i.e., you define a line-charge density ##\lambda(\vec{x})## along the wire. The dimension of this quantity obviously is ##\mathrm{C}/\mathrm{m}##.
Now for all these kinds of "singular" charge distributions you can again define a current-density vector, i.e., the surface-current-density vector ##\vec{j}_{\sigma}=\sigma \vec{v}## or line-current-density vector ##\vec{j}_{\lambda} = \lambda \vec{v}##. In both cases ##\vec{v}=\vec{v}(t,\vec{x})## is the velocity field of the charges along the surface or line, respectively. You can easily determine the dimensions of these quantities now yourself!
