What is a singleton in C#?

It's a design pattern and it's not specific to c#. More about it all over the internet and SO, like on this wikipedia article.

In software engineering, the singleton pattern is a design pattern that is used to restrict instantiation of a class to one object. This is useful when exactly one object is needed to coordinate actions across the system. The concept is sometimes generalized to systems that operate more efficiently when only one object exists, or that restrict the instantiation to a certain number of objects (say, five). Some consider it an anti-pattern, judging that it is overused, introduces unnecessary limitations in situations where a sole instance of a class is not actually required, and introduces global state into an application.

You should use it if you want a class that can only be instanciated once.