The static keyword, when applied to a class, tells the compiler to create a single instance of that class. It is not then possible to 'new' one or more instance of the class. All methods in a static class must themselves be declared static.
It is possible, And often desirable, to have static methods of a non-static class. For example a factory method when creates an instance of another class is often declared static as this means that a particular instance of the class containing the factor method is not required.
For a good explanation of how, when and where see MSDN