Relationship between C#, .NET, ASP, ASP.NET etc

The key points to take into account:

1. C# and C#.NET are the same thing... C#
2. .NET is, as you say, a library of code that .NET languages can talk to.
3. .NET languages come in different flavors such as: C#.NET, VB.NET, Managed C++, F#.
4. .NET languages compile to CIL ([Common Intermediate Language][1]) which means they all start "talking" the same language and can therefore inter operate.
5. ASP.NET is the portion of the .NET library used for making web sites. There are other subsections of ASP.NET like WebForms (the old way of making web pages) or the rapidly maturing MVC library that are worth looking at too.
6. Forms (old tech) or the new WPF (Windows Presentation Foundation) are the technologies you'd typically use in .NET to create what you know as traditional desktop applications.

One final thing I'd like to finish on is the difference between library and framework. In recent years these two terms have been used as those synonymous, however that is not the case. The easiest way I can think to differentiate the two is:

• A library contains many pieces of functionality that you may pick and choose from i.e. using one piece of technology doesn't mean you're locked into the rest. This means freedom, however you will have more work cut out for you.
• A framework however very much sets out how you will be working be providing a workflow that for better or worse is hard to change. This means rapid development/prototyping, but if significant changes are made in the future it may be impossible (or very time consuming) to implement them.

The project you're working on will depend on which choice you make.