This sort of question comes up ever so often so I thought I'd write a generic answer that can be given out, it's not perfect but here goes:
Re-introducing the Amiga
So, you've been away and want to know what's been happening in the Amiga scene?
If so this is for you.
The original Amiga was based on a Motorola 68K processor and a set of custom chips for graphics, sound etc.
Machines based on this technology have not been manufactured since 1996 and things have moved on since then.
As with any popular platform the users have decided to continue with it when the company went bust or did something incredibly stupid (unfortunately both common occurrences in the Amiga world). For those wanting to get the feeling back there are now several ways of running old and new Amiga software, these range from emulators to not one but two complete platforms.
If you intend to run old A500 / A1200 games you'll need an emulator such as UAE / Amiga Forever. The other systems can only run Amiga software which does not require the custom chips.
UAE
UAE is an emulator which runs on a variety of different platforms including Windows (WinUAE).
This emulates various types of Amiga including all the old favourites and even includes things like CD32.
You'll need to get hold of the Amiga OS for this. There are various ways of doing this which usually involve having an original Amiga but your best bet is to just to get the commercial version.
Cloanto's Amiga Forever
This is the said commercial version. It includes WinUAE, Amiga ROM and disc files and a whole heap of other stuff besides (extensions, applications, games). It's cheap, easy to use and well recommended.
AROS
AROS (Amiga Research Operating System) is a portable open source OS designed to be compatible with AmigaOS 3.1.
It is designed to be source compatible rather than binary compatible on different platforms so it wont run your amiga favourites on the PC.
New Platforms
There are two entire platforms which have updated the Amiga experience to modern PowerPC CPUs and standard PC expansion cards, neither can natively run orignal titles which require the custom hardware but can solve that with UAE.
AmigaOS 4.0 / AmigaONE (The Red pill)
This is the "official" solution sanctioned by the AmigaOS copyright owners.
AmigaOS 4.0 is a PowerPC native version of the AmigaOS, it is (at time of writing) available as a pre-release, it only runs on the AmigaONE PowerPC motherboard.
There are plans to make it available for Amiga PowerPC cards.
The AmigaONE also support PowerPC Linux.
MorphOS / Pegasos (The Blue Pill)
MorphOS is a PowerPC native OS developed by long time 3rd party Amiga developers.
It can run binaries for AmigaOS 3.x as well as those which require 3rd party extensions.
It currently runs on the Pegasos PowerPC motherboard though an old (limited) beta will run on Amiga PowerPC cards.
There are plans to get it running on Mac systems and possibly on the PC via the PearPC PowerPC emulator.
The Pegasos also support PowerPC Linux and many other Operating Systems are in development.
eBay
You could of course just buy an original.
And finally
The Amiga on-line boards can at times be subject to big flame wars.
Most Amiga users are reasonable prople and get long fine, unfortunately the Amiga community also has a small number of very loud idiots. While this is true of any on-line community the idiots are for some reason louder in the Amiga community.