BB1 games only run on BB1.
Some BB2 games will run on a BB1 (typically the earlier ones that don't use dongles and do not require a widescreen monitor). BB2 dongle games only run on a bb2.
BB3 games (also called blade games because the bb3 cabinet is called a blade cabinet), only run on a bb3 mpu.
I would say that the most versatile cabinet is a bb2 cabinet, since you can convert it into a transmissive reel cabinet very easily and be able to play transmissive reel games which are a lot of fun. You can also put a bb3 mpu in a bb2 cabinet, you just need an emotive controller (unless you get a bb2e cabinet that already has the emotive stuff) and an auxilliary power harness for it to work, and can run well over 90% of blade games (there are a handful of games that specifically require the higher resolution displays, but only a handful).
Blade cabinet are kind of expensive still, although they look really cool, but converting them to run anything other than video is difficult.
You can also run bb1 games on a bb2, you will need a standard button panel, a bb1 mpu, button harness to go from backplane to button panel, and a vga cable to connect the main monitor.
Some folks have upgraded their bb1 to bb2 by replacing the monitors (some modification to the door is required as well as a creative way to mount a top 22 inch monitor inside the bb1 tall box), but with the price difference now between a bb1 and abb2 cabinet being much lower, it is no longer recommended unless you already have a bb1 cabinet.