The short list of languages supported by Microsoft's .NET Framework is C++, JScript, VBScript, J#, and C#. C# is a language designed specifically for .NET, and most effectively takes advantage of its features. The other languages have been more or less retrofitted with extensions that allow them to take advantage of .NET's features. C++ specifically has the ability to bridge the gap with native code, and it's most commonly used as a middle layer inbetween .NET apps and native code. Also available are Python and Ruby, and people are working on some unholy unions like L#.
At first blush, .NET looks a lot like Java: sources are compiled to machine-independent bytecode run through a JIT compiler to achieve acceptable speed. .NET improves upon this idea by separating the runtime from the language. The Common Language Runtime (CLR) manages security and interoperability of programs and libraries. All of the languages mentioned above are compiled into IL (Intermediate Language), allowing developers to write code in whatever they're most familiar with and integrate output from multiple languages into a single project. Furthermore, plenty of wrappers around traditional C/C++ libraries exist (you can spot them because they typically have a # appended to the name), so you can get up to speed quickly.
But a cross-platform environment would be no good if everything only ran on Windows. Thankfully, Microsoft has made some of the core parts of .NET (the CLR and C#) ECMA standards, freely available for everyone to use. The folks over at Mono are working on ".NET for the rest of us," and so far have it working on Linux, Solaris, OS X, and Windows. In fact, if you're running Ubuntu, you probably already have it installed.
So why bother? Well, first check out swarm.NET. It's a C# port of swarm.3 as seen on my hacks page, using SDL. That was written and debugged on my laptop, running Slackware 12. The resulting executable runs without modification on Windows. That is, I can make decently-performing games that run on every major platform out there, out of the box, without ever touching a Windows machine. That is hot shit.
I think Nancy put it best: "This sounds like the way it's meant to be..."