Late answer, but I figured it's worth keeping this updated. Expanding on Andrea Turri answer to reflect updated jQuery API and .success/.error deprecated methods.
As of jQuery 1.8.* the preferred way of doing this is to use .done() and .fail(). Jquery Docs
e.g.
$('#my_get_related_keywords').click(function() {
var ajaxRequest = $.ajax({
type: "POST",
url: "HERE PUT THE PATH OF YOUR SERVICE OR PAGE",
data: '{"HERE YOU CAN PUT DATA TO PASS AT THE SERVICE"}',
contentType: "application/json; charset=utf-8",
dataType: "json"});
//When the request successfully finished, execute passed in function
ajaxRequest.done(function(msg){
//do something
});
//When the request failed, execute the passed in function
ajaxRequest.fail(function(jqXHR, status){
//do something else
});
});