Ok...
The Amiga
was a series of computers released in the eighties/nineties. Many people like its hardware architecture, it was ground-breaking at the time. Many people like the OS (previously called Workbench, now AmigaOS) because it is a low-footprint, fast, easy to configure, efficient OS.
Amigas were used primarily for gaming, but also for professional graphics work, notably TV editing, as well as standard office'y type applications stuff.
Here is a good guide to the history of the Amiga.
Here is a guide to the evolution of Workbench/AmigaOS.
Currently what is "Amiga" is a bit fragmented. AmigaOS 4.0 is due to arrive "at some point soon", MorphOS is an OS available now that is applications-compatible with the old range of Amigas, and AROS is an x86 port of AmigaOS.
What can be done with an Amiga? 99% of the stuff you can do on a PC, though it lacks commercial support, which means that most of the applications available are freeware, and so there are little or no big projects to write Amiga applications.