PHP

function myFunction(){
	//do something

	//echo json_encode(array); // if required json result
	die();
}
add_action('wp_ajax_myFunction', 'myFunction');
add_action('wp_ajax_nopriv_myFunction', 'myFunction');

-------------------------------------------------------------------

Jquery

jQuery(document).ready(function($) {

	 $.ajax({
	 	url: ajaxurl,
		data: {
	 			'action':'myFunction',
	 			'data1' : 'value1',
	 			'data2' : 'value2'

	 	},

		// dataType: 'JSON', // if required json result

	 	success:function(data) {
	 		//var obj = jQuery.parseJSON(data); // converting string to json object
	 		console.log(data);
	 	},

	 	error: function(errorThrown){

			 console.log(errorThrown);
	 	}

	 }); 

});

http://wptheming.com/2013/07/simple-ajax-example/

http://codex.wordpress.org/AJAX_in_Plugins