When should I use a semicolon after curly braces?

You use a semicolon after a statement. This is a statement:

var foo = function() {
  alert("bar");
};

because it is a variable assignment (i.e. creating and assigning an anonymous function to a variable).

The two things that spring to mind that aren’t statements are function declarations:

function foo() {
  alert("bar");
}

and blocks:

{
  alert("foo");
}

Note: that same block construct without semi-colon also applies to for, do and while loops.