How can I find out which method is best for a situation? Can anybody provide some examples to know the difference in terms of functionality and performance?
Each one of them uses XMLHttpRequest. This is what the browser uses to make the request. jQuery is just a JavaScript library and the $.ajax method is used to make a XMLHttpRequest.
$.post and $.get are just shorthand versions of $.ajax. They do pretty much the same thing but makes it quicker to write an AJAX request - $.post makes a HTTP POST request and $.get makes a HTTP GET request.