Here's a good best practice way to do things. There are sometimes reasons to violate these rules, but this will usually get you a pretty optimal setup.
You have two "grounds" in your cabinet. One is Earth ground (aka "Frame", aka "Safety Ground", aka "The third prong on your AC plug"); it's usually green or green with a yellow stripe, but you'll also frequently see big braids. The other is "power common" (aka "negative", aka "DC-", aka "ground") and is conventionally black. So, the question is what to do with them.
Earth ground is pretty simple. Hook everything that is exposed metal but otherwise not power related to it. This would be things like your coin door, CP hinges, exposed bolts, etc. This prevents static buildup and also serves a safety purpose of catching any electrical faults inside the cabinet by forcing the exposed metal to be "safe" to the touch. Ideally you'd run big-honkin wires from each metal part to a single point, but it's also fine to just daisy chain metal chunks one to the next.
Power common is used for power (duh), signal common (like the common ground bus on your CP switches), and other DC power related things. You'd use this when running power to DC devices like DC coin lamps, button LEDs, keyboard encoders, etc. and for the return/reference on I/O like switches, trackball signals, etc.
You shouldn't mix them together all willy nilly, but they do tend to get connected together at your monitor: your monitor has a signal coming in to it for the video but also has exposed metal parts that get grounded to the earth ground. This "weak connection" (it's a physically large loop) can result in interference patterns. If this happens, try connecting the grounds together with a good solid connection at one point. Conventionally, this is right at the DC power supply. For you PC users, this is already done as all PC supplies do this internally.
The reason for all this separation is that unexpectedly large currents can flow on the earth ground network in order to keep everything at the same potential (voltage): this is what prevents static buildup. The side effect of this is that "ground" isn't always quite the same on that earth ground network due to line loss caused by these currents. Remember that voltage is all relative (that's why your multimeter has two probes), so the voltage difference seen at various points in the cabinet would differ simply because ground isn't always at the same potential. Separating power/signal ground from earth ground fixes this problem. The reason that tying your grounds together with a good solid connection at the power supply can fix monitor interference issues is precisely this reason: all those currents will flow over that "solid" connection rather than going over your monitor wiring (which is a relatively "weak" path).
It's also best to try to avoid mixing high current loops (like power runs for your game boards) with low current signals (like button returns) on your DC power common for similar reasons as mentioned above. For example, running your video ground straight back to your power supply rather than using the pin on JAMMA would not be advised (this is in fact one reason why JAMMA has a dedicated video ground pin).
Now, all this is the conventional "best practice". These rules get violated all the time, and usually it's no big deal. However, if you're having ground related issues like coin doors that static zap you or scrolling horizontal black bars on your monitor, you may want to check out your ground configuration and see if it can be improved.