I found out about FS-UAE thanks to a post on Binary Devotion, so thanks guys! This then led me to the official website of the emulator. Don't worry - I'm keeping a list of the URLs in the "Useful links" section to the left, so they will be within easy reach should you lose track of this post.
FS-UAE, like any Amiga emulator, requires a Kickstart ROM file in order to boot-up into a working Amiga system. It does come with an emulated AROS ROM, but I don't know whether this will work with games.
Luckily for me, I have a few Amigas available to nab the ROM file from - my A1200 provided the Kickstart 3.1 (v40.68) file I'm using. Various flavours are available with the Amiga Forever pack from Cloanto - not all games will work with the later Kickstarts, so you may need Kickstart 1.3 for particularly old games.
The readme file which comes with the MacOS distribution doesn't say much about where it should or shouldn't reside, so I've installed the FS-UAE folder with the /Applications/ folder on the main disk. Running the FS-UAE.app file straight away opens a window telling me that there isn't a valid configuration file. Guess what I'll be working on next!
It does, however, boot to an AROS screen saying "No valid media". To close the window, you need to F12 - in order to do this in MacOS X, you'll need to set the function keys as standard function keys.