Q:

Static Class and Private Constructor

I am not able to understand this. I tried doing a small example in VS2010 as below.
"Creating a static class is therefore much the same as creating a class that contains only static members and a private constructor"
Does this statement means a class with private constructor and one or more static methods like below is static class ? I know we call a class static only when static keyword is mentioned in class. 
Also, we cannot inherit the below class and also we can't instantiate this class right ? 
public class Base
{
    private Base() { Console.WriteLine(" I am from normal Base constructor"); }
    static void NewMethod() { Console.WriteLine("Hey I am from Static Base"); }
    public void New() { } 
}

A:

Creating a static class is therefore much the same as creating a class that contains only static members and a private constructor

This statement is attempting to get across the right idea but failing to do so. A static class is not like a class with only static members and a private constructor. Here's a class with static members and private constructor:
class X
{
    private X() {}
    public static X Y() { return new X(); }
}

But that's not at all like a static class! The author of that statement seems to think that having a private constructor prevents you from making instances, but of course it does not. 
A more accurate statement would be:

Creating a static class is much like creating a class that is both abstract and sealed, and contains no instance members. 

And in fact, when the C# compiler generates the code for a static class, that's precisely what it does: it marks the class as both abstract (so it cannot be instantiated directly) and sealed (so that it cannot be extended). 
I note that it is not legal for you to declare a class both abstract and sealed yourself; the only way to do so in C# is to make a static class.
I will bring the misleading sentence to the attention of the MSDN documentation managers. Thanks for pointing it out.

A:

What it means is that:
public static class Foo
{
    public static void Bar() { }
}

is essentially the same as
public class Foo
{
    private Foo() { }
    public static void Bar() { }
}

because if the class only has a private constructor, it cannot be instantiated outside the class.

A:

"Does this statement mean a class with private constructor and one or
  more static methods like below is a static class?"

The answer is No, and one difference is explained in exactly the next sentence after the one you are citing from MSDN:

The advantage of using a static class is that the compiler can check
  to make sure that no instance members are accidentally added.

It means that you will get a compiler error in Class2 shown below.
public class Class1
{
    private Class1() { }
    public static void Method() { }
    private string member; // valid, but pointless
}

public static class Class2
{
    public static void Method() { }
    private string member; // error CS0708
}

More important, although Class1 has a private constructor, it may still be instantiated:
public class Class1
{
    private Class1() { }

    private static Class1 instance = new Class1();

    public static Class1 Instance
    {
        get { return instance; }
    }
}

A static class on the other hand, may never be instantiated.

