Start out by watching this video and try to gleam the basics of what a capacitor is.
Here is a simple diagram of how capacitors function.
Axial simply referes to the terminal leads being located at the top and bottom of the capacitor.
Axial capacitors can be polarized, non-polarized, and bi-polarized. Polarized will be the easiest to find, plus the cheapest. But you have to take care when soldering the terminal leads to the coils. If you get the + switched with the – the capacitor will short out and fail completely. You can tell which end is + or – by the markings along the casing. Another way to tell is that the + terminal lead will be slightly longer than the negative terminal lead.
Non-polarized axial capacitors can be wired to the coil with the +/- terminals backwards as this is pretty much what non-polarized means.
The voltage rating is simply how high a voltage the capacitor can take before it shorts out and fails. 25v-50v should be fine for most machines. Unless you plan on running your machine at over 25 volts of power.
The uf rating is the most important in regard to tattoo machine set up. Tattoo machines are commonly set up with 22uf for liners, 33uf for custom set ups, and 47uf for shaders/color machines. Experiment with different capacitors and see what effects you can produce.
“The circuit created when you combine a capacitor with the magnetic coils is called a Resonant Tank Circuit. It essentially alternately stores energy in the field of the coils and between the plates of the capacitor. Thus it is creating AC from the DC current. The frequency at which the tank circuit runs is determined by the combination of the coils and the capacitor, change the value of either of them will change the operation frequency of your needle.” -From physicsforums.com