Singleton class is a type of class which can have only one of it object through out the application. It does not allows its object to be created by the calling code e.g Class obj = new Class;
Instead it have a private constructor so that any other code can't create an object of class and it exposes a static method which internally creates an object of class if not already created or creates it and return to the calling code.
Singleton class is mainly used for maintaining some information which would remain same throught out the application. Some of the example are Maintaining Connectionstring, Registry values in a object etc.
If you want to know more you can go through dofactory.com