Can the .NET framework be installed on Linux, Unix, MAC etc. operating systems?
.Net can be looked at as two separate parts. One is the ECMA standardized CLI and the other is CLR. CLR is Microsofts implementation of CLI.
CLI can be executed on many devices, such as Windows (x86, x64, ia64 and now ARM), XBox, Zune, mobile phones and even embedded devices (small chips). Mono/Moonlight are implementations of CLI and even parts of CLR partially funded by Microsoft through Novell. Microsoft supports this work and releases source code to aid. Through Mono you can reach most operating systems and CPU types.
There are also compilers that can compile .Net code directly to platform independent code that can be directly executed by the CPU. Nice in case you want to write your own operating system in C#. (I don't remember names, but I can Google some links if interested.)
CLI is designed to be platform and langugage independent. It runs well on most operating systems and CPU's through various implementations. In practice does this mean that it supports for example language features such as case sensitive/case insensitive variables (+ muuuch more) and it has an instruction set compatible with most CPU's (the CLI machine code resembles the instructions most CPU's offer).
Implementations of .Net (overview to answer your question):