What is the exact definition of a closure?

No, that's not a closure. Your example is simply a function that returns the result of incrementing a static variable.

Here's how a closure would work:

function makeCounter( int x )
{
  return int counter() {
    return x++;
  }
}

c = makeCounter( 3 );
printf( "%d" c() ); => 4
printf( "%d" c() ); => 5
d = makeCounter( 0 );
printf( "%d" d() ); => 1
printf( "%d" c() ); => 6

In other words, different invocations of makeCounter() produce different functions with their own binding of variables in their lexical environment that they have "closed over".

Edit: I think examples like this make closures easier to understand than definitions, but if you want a definition I'd say, "A closure is a combination of a function and an environment. The environment contains the variables that are defined in the function as well as those that are visible to the function when it was created. These variables must remain available to the function as long as the function exists."

For the exact definition, I suggest looking at its Wikipedia entry. It's especially good. I just want to clarify it with an example.

Assume this C# code snippet (that's supposed to perform an AND search in a list):

List<string> list = new List<string> { "hello world", "goodbye world" };
IEnumerable<string> filteredList = list;
var keywords = new [] { "hello", "world" };
foreach (var keyword in keywords)
    filteredList = filteredList.Where(item => item.Contains(keyword));

foreach (var s in filteredList)  // closure is called here
    Console.WriteLine(s);

It's a common pitfall in C# to do something like that. If you look at the lambda expression inside Where, you'll see that it defines a function that it's behavior depends on the value of a variable at its definition site. It's like passing a variable itself to the function, rather than the value of that variable. Effectively, when this closure is called, it retrieves the value of keyword variable at that time. The result of this sample is very interesting. It prints out both "hello world" and "goodbye world", which is not what we wanted. What happened? As I said above, the function we declared with the lambda expression is a closure over keyword variable so this is what happens:

filteredList = filteredList.Where(item => item.Contains(keyword))
                           .Where(item => item.Contains(keyword)); 

and at the time of closure execution, keyword has the value "world," so we're basically filtering the list a couple times with the same keyword. The solution is:

foreach (var keyword in keywords) {
    var temporaryVariable = keyword;
    filteredList = filteredList.Where(item => item.Contains(temporaryVariable));
}

Since temporaryVariable is scoped to the body of the foreach loop, in every iteration, it is a different variable. In effect, each closure will bind to a distinct variable (those are different instances of temporaryVariable at each iteration). This time, it'll give the correct results ("hello world"):

filteredList = filteredList.Where(item => item.Contains(temporaryVariable_1))
                           .Where(item => item.Contains(temporaryVariable_2));

in which temporaryVariable_1 has the value of "hello" and temporaryVariable_2 has the value "world" at the time of closure execution.

Note that the closures have caused an extension to the lifetime of variables (their life were supposed to end after each iteration of the loop). This is also an important side effect of closures.

From what I understand a closure also has to have access to the variables in the calling context. Closures are usually associated with functional programming. Languages can have elements from different types of programming perspectives, functional, procedural, imperative, declarative, etc. They get their name from being closed over a specified context. They may also have lexical binding in that they can reference the specified context with the same names that are used in that context. Your example has no reference to any other context but a global static one.

From Wikipedia

A closure closes over the free variables (variables which are not local variables)

A closure is an implementation technique for representing procedures/functions with local state. One way to implement closures is described in SICP. I will present the gist of it, anyway.

All expressions, including functions are evaluated in an environement, An environment is a sequence of frames. A frame maps variable names to values. Each frame also has a pointer to it's enclosing environment. A function is evaluated in a new environment with a frame containing bindings for it's arguments. Now let us look at the following interesting scenario. Imagine that we have a function called accumulator, which when evaluated, will return another function:

// This is some C like language that has first class functions and closures.
function accumulator(counter) {
    return (function() { return ++counter; });
}

What will happen when we evaluate the following line?

accum1 = accumulator(0);

First a new environment is created and an integer object (for counter) is bound to 0 in it's first frame. The returned value, which is a new function, is bound in the global environment. Usually the new environment will be garbage collected once the function evaluation is over. Here that will not happen. accum1 is holding a reference to it, as it needs access to the variable counter. When accum1 is called, it will increment the value of counter in the referenced environment. Now we can call accum1 a function with local state or a closure.

I have described a few practical uses of closures at my blog http://vijaymathew.wordpress.com. (See the posts "Dangerous designs" and "On Message-Passing").

There's a lot of answers already, but I'll add another one anyone...

Closures aren't unique to functional languages. They occur in Pascal (and family), for instance, which has nested procedures. Standard C doesn't have them (yet), but IIRC there is a GCC extension.

The basic issue is that a nested procedure may refer to variables defined in it's parent. Furthermore, the parent may return a reference to the nested procedure to its caller.

The nested procedure still refers to variables that were local to the parent - specifically to the values those variables had when the line making the function-reference was executed - even though those variables no longer exist as the parent has exited.

The issue even occurs if the procedure is never returned from the parent - different references to the nested procedure constructed at different times may be using different past values of the same variables.

The resolution to this is that when the nested function is referenced, it is packaged up in a "closure" containing the variable values it needs for later.

A Python lambda is a simple functional-style example...

def parent () :
  a = "hello"
  return (lamda : a)

funcref = parent ()
print funcref ()

My Pythons a bit rusty, but I think that's right. The point is that the nested function (the lambda) is still referring to the value of the local variable a even though parent has exited when it is called. The function needs somewhere to preserve that value until it's needed, and that place is called a closure.

A closure is a bit like an implicit set of parameters.