You can kind of equate the .NET framework to Java--it is a collection of classes and data types built by Microsoft that allow developers to create full applications in short amount of time (when compared to a language like C/C++). In Java terms, it is the API.
If you are not of a programmer background, then think of the framework as a set of softwares (that you would not run directly) that other softwares you run rely on. An example you may be familiar with is a program that has many .dll files. These dll files contain code that the application you are running relies on. If you delete these files, your application will not run. The same can be said of the framework (and in fact, the framework is a collection of dll's)--if you delete/remove the framework, some of your applications will no longer run.
Many of MS' newer applications leverage the framework--Windows 7 has parts that rely on it, I believe. This is the reason you are being prompted to install it.
The current version in production is 3.5, as Patmac951 pointed out. However, 4.0 is scheduled to be released some time this year.
