Arcade games use one of two types of power supplies, either a switching power supply (much like modern computers use), or a linear power supply (used mostly in older games).
An arcade switching power supply looks similar to a small car audio amplifier, these are often referred to as "switchers". It will have 2 screws for hookup to AC power, and then a row of screws for the DC power it puts out. The vast majority of these power supplies are capable of putting out +5, +12, and -5 volts (most games do not use the -5 volts at all). These units are self contained, and require little maintenance (they do have a little knob on the side that you can adjust them with as needed). Defective switching power supplies can be cheaply repaired by a competent technician (although the industry standard is just to throw them away). You can replace an arcade switching power supply with a standard PC power supply in a pinch (but a few games want more out of the -5 volts setting than the PC power supply can put out).
Now a linear power supply is a whole different kind of beast (most older games like Ms. Pac-Man and Turbo use a linear power supply). Linear power supplies vary wildly in capabilities and in construction, but they all have a few things in common. Inside they contain a transformer, rectifier, and smoothing capacitor. The exact amount and types of DC current they put out is quite different from game to game (they don't have a set standard like the switching supplies do). Linear power supplies have a much higher failure rate (it seems even higher due to the fact that most of them are almost 20 years old now), than switching power supplies do. But fortunately they are not very difficult to repair (due to their simplicity).
The main difference between the two types (as far as normal people are concerned), is that a linear supply can operate without any load at all (while the switching units will fry themselves if left on without anything attached). The switching supplies require almost no adjustment ever, while the linear ones need adjustment fairly frequestly (but they do put out cleaner power when properly tuned).