jQuery.load() is probably the easiest way to load data asynchronously using a selector, but you can also use any of the jquery ajax methods (get, post, getJSON, ajax, etc.)
Note that load allows you to use a selector to specify what piece of the loaded script you want to load, as in
$("#mydiv").load(location.href + " #mydiv");
Note that this technically does load the whole page and jquery removes everything but what you have selected, but that's all done internally.