The .NET Framework class library is exactly what its name suggests: a library of classes and other types that developers can use to make their lives easier. While these classes are themselves written in C#, they can be used from any CLR based language.
The Framework Class Library (FCL) is the wider library that contains the totality: ASP.NET, WinForms, the XML stack, ADO.NET and more. You could say that the FCL includes the BCL.
On a simple level, .NET Framework = libraries (FCL, BCL), language compilers (C#, VB.NET) and Common Language Runtime (CLR).