Thomas's reply was bang on. If a bit technical
The amiga A500 hardware could display 16 or 32 colours in game. The copper could be used to change colours per line for all those lovely fades you see in amiga games
It also had a 4096 HAM mode (mainly for static imagery- Art programs, adventure games, loading screens etc). You could also have different colour/resolution screens dragged up for menu bars etc. I used have a ham hires mod player that looked like a stereo- it used to sit at the bottom of a 16 clour lo-res workbench
The the A500+ and A600 EHB mode was added allowing 64 colours (darker versions of the first 32 colours)
This is all in hardware: coders produced their own code for introducing more colours: Lion heart had hundreds of colours on screen, as did Universe (256 from 4096) rumour has it they picked certain ham colours that caused no fringing, and just diplayed them.
-EDIT- Iv'e just realised mine sounds technical as well, guess there is no easy way to explain it
