The second example simply defines an instance of Object, with a few properties set. The first one is much more interesting.
Functions in JavaScript have dual purposes, one as a normal function/procedure, and the other as the basis for prototypal inheritance, similar to a "class" in normal OOP.
For example:
var apple = function() {
this.type = 'granny smith';
};
var myApple = new apple();
alert(myApple.type); // -> 'granny smith'
Here, we've defined a "class" called apple, with an initializer that sets the type property. myApple is then created as an instance of apple, and the type property gets set by the apple() initializer.
(As an aside, new apple() could also be called as new apple with the parenthesis omitted and the result is still the same, the initializer is still called, it just receives no arguments)
The other major thing that using new gives us is prototypal inheritance. All functions come with a prototype object that we can set properties on. Those properties then become fallback values for instance objects that don't have their own value defined.
Example:
var greenApple = function() { };
var myApple = new apple();
alert(myApple.color); // -> undefined
greenApple.prototype.color = 'green'
alert(myApple.color); // -> green
Here, setting greenApple.prototype.color affected the color property for the myApple we had already created. This is what is referred to as prototypal inheritance, and it has a wide variety of uses.
Edit: The following refers to an earlier version of the question that was edited while I was writing this. You can safely ignore it unless you're especially interested.
Now, back to your first example:
var apple = new function() { ... }
Because you have the new keyword before the function definition, the effect is like creating the apple class, making a myApple object, then throwing the apple class away. If you had access to this class, you could use .prototype to add prototype properties, but you don't - all that's contained in the apple variable (in OOP terms) is an instance of the class, not the class itself.