Mono is an implementation of the Common Language Runtime and some parts of the .NET Framework that runs on Linux, BSD, MacOSX, Windows amongst others (Here you can see the supported platforms). The idea is to allow code developed for .NET to be used on as many platforms as possible.
Mono is not an IDE, but there is an IDE that runs on Mono, the IDE itself it’s called MonoDevelop.
Cross-platform means that you can compile once and run it in multiple platforms, in this case, compile once and run in every platform supported by Mono.
.NET supports many different languages (C#, C++, VB.NET, Fortran, Perl, Python amongst others), but the most common ones are C# an VB.NET