Be prepared to be bombarded with information from the three current contenders to the Amiga throne... AmigaOS, MorphOS and AROS. Quick summary, I'm sure the intersted parties will provide more details...
1. AmigaOS (running on the AmigaOne): PowerPC operating system, will not run on a pure M68K machine. Still in closed beta, supposedly hitting public beta this December. Written by Hyperion Entertainment, based off original AmigaOS source code (but largely a complete rewrite - all the BCPL and m68K asm got rewritten in C and some PPC asm), so it works internally very similarly to 3.x. The AmigaOne it runs on is a modified Teron PX, arranged by Eyetech. It's source-compatible with AmigaOS 3.x and will run 68K binaries via emulation. Will also run on Cyberstorm and BPPC accelerators.
2. MorphOS (running on the Pegasos): Another pure PPC OS, built as a "box" (basically a protected subsystem) on top of a microkernel. Written by Genesi (same people who sell the Pegasos). Out of beta for about two or three versions now (it's in 1.4 now, but I believe it was a beta up until 1.2, maybe 1.1). Unfortunately only runs on a machine that is currently no longer in production (the Pegasos 1), but the Pegasos 2 will supposedly start selling sometime this December, and you should be able to buy a 2nd hand Pegasos 1. It's source-compatible with AmigaOS 3.x and will run 68K binaries via emulation. Will also run on some PPC cards, but I can't remember which ATM.
3. AROS (running on an x86 PC or classic Amiga): The open-source alternative, by Team AROS. Shares some code with MorphOS, but it's a lot closer to AmigaOS architecturally. It's source-compatible with AmigaOS 3.x but AFAIK they don't have an emulator so you can't run old programs unless you're using a real Amiga.
Neither of the three have chipset emulation so you can't run old games on either - you must use something like UAE for that. For system-legal apps, you should be OK. In terms of price, the AmigaOne is more expensive ($750-$850 for a G4 model, depending where you buy it), the Pegasos is cheaper ($725 for a G4 Pegasos2, around $300 for a G4 Pegasos 1, if you can find it). Cheapest is AROS (runs on the PC you already have :-))
Here's the rub: there's plenty of animosity between the three groups (rather, between followers of AmigaOS on the one side ("red") and the other two on the other ("blue" for MorphOS, I have no idea if AROS has picked a colour...). The reasons are mostly due to historical disputes, there's no real reason why they could not be cooperating. So if anyone starts making extreme claims about any solution (along the lines of "it will never be released" or "it's based on stolen code") just treat it with a large pinch of salt.
This animosity also means that AmigaOS won't run on a Pegasos, and MorphOS won't run on an AmigaOne. There are people who say they got it to work, and they probably did, but these are not officially supported options (again, those disputes).
