Using Low Search for ExpressionEngine with Ajax. Change any of the three variables on top to suit your needs. Tested with jQuery 1.9.0.

low-search-ajax.js

(function($){
	$(function(){
		var $form   = $('#search'), // Search form
		    $target = $('#results'), // Results container
		    rp      = 'search/ajax-results'; // Template for results only

		// Function to execute on success
		var success = function(data, status, xhr) {
			$target.html(data);
		};

		// Hijack the form on submit
		$form.submit(function(){

			// Tell target we're loading
			$target.html('<p>Loading...</p>');

			// Add custom result page to data
			var data = $form.serialize()
			         + '&result_page=' + rp;

			// Perform the ajax call
			$.ajax({
				type: 'POST',
				url: $form.attr('action'),
				data: data,
				success: success,
				dataType: 'html'
			});

			// Don't submit the form afterwards
			return false;
		});
	});
})(jQuery);

This approach still works today, but you might want to tweak the jQuery syntax (or use some other JS lib).