html - jQuery click function doesn't work after ajax call?

Since the class is added dynamically, you need to use event delegation to register the event handler like:

$('#LangTable').on('click', '.deletelanguage', function(event) {
    event.preventDefault();
    alert("success");
});

This will attach your event to any anchors within the #LangTable element,
reducing the scope of having to check the whole document element tree and increasing efficiency.

FIDDLE DEMO