There are only really two main (free) Amiga emulators. From experience, I’ve found that UAE has the most features and highest compatibility. There is also Fellow, if UAE isn’t for you.
UAE is a portable emulator, with flavours for pretty much all operating systems. On Windows XP, I use WinUAE, as it very simple to use with a great GUI. On my Mac, I use E-UAE, which is based on the same code but doesn’t yet have an Intel binary so isn’t as fast or smoothly running as it could be.
UAE can emulate several different processors and allows many different configurations for memory, OS, etc. It can even access your host computer’s devices (even the CDROM and modem) as Amiga drives from within the emulator!
You will also need OS roms and may need Workbench disk images to run UAE. The emulator comes with tools you can use on a real Amiga to extract images of its roms, which you can transfer to the emulator. Similarly, tools are available to make images of your real Amiga disks.
Alternatively, you can buy Amiga Forever. It’s a compilation of Amiga stuff – emulators (UAE and Fellow), legal copies of Workbench and Kickstart roms, piles of demos, games and applications, with everything ready configured. It’s the easiest way to get an Amiga up and running on your current computer.
UAE is absolutely essential if you’re interested in Amigas. I’ve even used it to recreate my real A1200,doing this to retrieve everything from the hard drive.