WordPress has decent enough search functionality. However, if you are looking for more features like a jQuery based advanced Ajax search form for your blog, then, I have sorted 4 best WordPress plugins. These plugins are highly customizable and they have functionality like search suggestions, auto-complete, include and exclude from search, set search algorithm, set search performance, compatible with other popular WordPress plugins, responsive and adaptive, and translation ready.
1) Ajax Search: This is one of the best plugins for adding jQuery based Ajax search in a WordPress blog. By using this plugin you can enable search for posts, pages and custom post types. There are more than 45 themes search form themes available with this plugin. You can add any one which matches your WordPress theme.
Here are the features of Ajax Search plugin:
i) Search suggestions and auto-complete: Plugin automatically adds search suggestions and auto-complete when a user starts typing in search form. Search suggestions are displayed with thumbnails in a drop down section just below the search form . Size of thumbnails can be set from the plugin settings page. Search auto-complete feature can be set according to keywords from Google or by using search statistics. Search statistics are generated when users use the search form.
ii) Include and exclude from search: You have the option to include comments search, title search, content search, search in post excerpts and exclude certain posts and categories from the search. This can be configured as per your needs.
iii) Set search weight: From the plugin settings page, you can set the relevance value for search based on exact match weight and random match weight values. You have options to set title weight, content weight, excerpt weight, and terms weight.
iv) Search performance: Plugin uses caching functionality for improving search performance.
v) Widgets support: Search form can be added to your blog by using widgets. You can place these widgets in any sidebar of your blog.
vi) Plugin compatibility: This plugin is compatible with BuddyPress, Jigoshop, WooCommerce, WP-ecommerce, WordPress multisite and network install. In BuddyPress, you have options to search even the forums and in groups, private groups and in users. In eCommerce store you can search products.
vii) Responsive search form: Search form is fully responsive and can be used in smart phones and tablets.
viii) Translation ready: You can use this plugin in any language.
2) Instant search: This plugin provides instant search capabilities in your blog. There are more than 24 themes of search display boxes to choose from. It can also log the incoming search terms. Search form is responsive and adaptive to different devices.
3) Category search: By the help of this plugin you can do a category search in your blog. It automatically excludes irrelevant items in the search and shows you the most relevant results. In this plugin, you can include different taxonomies for search. Plugin generates HTML code for the search form which you can place anywhere in your site. It also generates a search widget for adding it in to a sidebar. This plugin adds an advanced search form in your WordPress blog. Users can search through both posts and custom post types. Search form can be added anywhere in the blog by using shortcodes.