Each computer family works with specifically tailored software, on the operating system level as well as on the application level. When you supply a computer with soft- and/or hardware so that it's able to use software written for a different platform, this is called emulation. The Amiga is sort of a world champion in emulating other platforms like the PC, Macintosh®, C64, Atari ST®, and more.
Not directly. In first line, the Amiga builds its own world with its own philosophy. Thanks to its flexibility, though, there are several ways to achieve MS-DOS® compatibility: With the current AmigaOS versions you can read and write disks directly in MS-DOS® format to grant easy data exchange. For low demands on processing speed there are software MS-DOS® emulators that fake a whole compatible PC purely in software and allow you to run standard DOS applications. For higher demands there are hardware emulators containing their own PC processor, thus offering full working speed.
Yes, there are emulators (like "Shapeshifter") available which transform the Amiga into an almost native Macintosh® without any loss of speed!