What is a singleton in C#?

What it is: A class for which there is just one, persistent instance across the lifetime of an application. See Singleton Pattern.

When you should use it: As little as possible. Only when you are absolutely certain that you need it. I'm reluctant to say "never", but there is usually a better alternative, such as Dependency Injection or simply a static class.

A singleton is a class which only allows one instance of itself to be created - and gives simple, easy access to said instance. The singleton premise is a pattern across software development.

There is a C# implementation "Implementing the Singleton Pattern in C#" covering most of what you need to know - including some good advice regarding thread safety.

To be honest, It's very rare that you need to implement a singleton - in my opinion it should be one of those things you should be aware of, even if it's not used too often.

another way to implement singleton in c#, i personally prefer this way because you can access the instance of the singeton class as a property instead of a method.

public class Singleton
    {
        private static Singleton instance;

        private Singleton() { }

        public static Singleton Instance
        {
            get
            {
                if (instance == null)
                    instance = new Singleton();
                return instance;
            }
        }

        //instance methods
    }

but well, as far as i know both ways are considered 'right' so it's just a thing of personal flavor.

You asked for C#. Trivial example:

public class Singleton
{
    private Singleton()
    {
        // Prevent outside instantiation
    }

    private static readonly Singleton _singleton = new Singleton();

    public static Singleton GetSingleton()
    {
        return _singleton;
    }
}