What is the (function() { } )() construct in JavaScript?

It's just an anonymous function that is executed right after it's created.

It's just as if you assigned it to a variable, and used it right after, only without the variable:

var f = function () {
};
f();

In jQuery there is a similar construct that you might be thinking of:

$(function(){
});

That is the short form of binding the ready event:

$(document).ready(function(){
});

But the above two constructs are not IIFEs.

It’s an Immediately-Invoked Function Expression, or IIFE for short. It executes immediately after it’s created.

It has nothing to do with any event-handler for any events (such as document.onload).
Consider the part within the first pair of parentheses: (function(){})();....it is a regular function expression. Then look at the last pair (function(){})();, this is normally added to an expression to call a function; in this case, our prior expression.

This pattern is often used when trying to avoid polluting the global namespace, because all the variables used inside the IIFE (like in any other normal function) are not visible outside its scope.
This is why, maybe, you confused this construction with an event-handler for window.onload, because it’s often used as this:

(function(){
  // all your code here
  var foo = function() {};
  window.onload = foo;
  // ...
})();
// foo is unreachable here (it’s undefined)

Correction suggested by Guffa:

The function is executed right after it's created, not after it is parsed. The entire script block is parsed before any code in it is executed. Also, parsing code doesn't automatically mean that it's executed, if for example the IIFE is inside a function then it won't be executed until the function is called.

Update Since this is a pretty popular topic, it's worth mentioning that IIFE's can also be written with ES6's arrow function (like Gajus has pointed out in a comment) :

((foo) => {
 // do something with foo here foo
})('foo value')