As VingtTrois said, the main difference in Amiga games is the chipset they require. Most games are designed for OCS systems, while others are either enhanced for AGA or written specifically and only for AGA Amigas. You will sometimes see mention of ECS, but this was only a slight enhancement over OCS which adds more screenmodes to Workbench but wasn't ever used by games.
OCS (Original Chip-Set) Amigas include:
A1000
A2000
A500
A1500
A2500
CDTV
ECS (Enhanced Chip-Set) Amigas include:
A500+
A3000
A600
A3000T
AGA (Advanced Graphics Architecture) Amigas include:
A1200
A4000
CD32
A4000T
Sometimes games will also require a specific Kickstart ROM, CPU, or a certain amount of RAM to run. Thankfully all Amigas can be upgraded to Kickstart 3.1, the latest ROM revision, and can also make use of ROM-switcher hardware so they can keep their original Kickstart (for compatibility or nostalgia) as well as use the best. A1000s don't have a Kickstart ROM, and need to read the Kickstart from a floppy disk unless they have been hacked/modified. You can also get accelerator cards which offer faster CPUs and more RAM for all Amigas (although some of these are as rare as hens' teeth!).