I've found that the concept of anonymous functions is the biggest hurdle to get over -- closures seem to follow as a natural extension of that.

I explained this to someone recently in the context of event handlers in ActionScript/JavaScript, starting with the idea that when you do:

  something.onclick = function(){...};

you're passing a function around just like it was any other value.

Then I moved to an artificial example of a 'magic function-making machine':

  function giveMeAnAdder(baseNumber) {
    return function (i) {
      return baseNumber + i;
    };
  }

with a couple of examples of storing and using the functions that giveMeAnAdder returns.

Then I pointed out that, normally, local variables in a function disappear when the function returns, so how does my adder 'remember' what baseNumber is? And the answer is that the anonymous function is actually a bundle of a function body and the local context, and we call this a closure.