WordPress already comes with a decent search functionality, but it has various limitations. Ivory Search is a plugin that lets you add a more advanced search to your site. You can use it to search images, files, attachments, and products. It integrates with bbPress, WooCommerce, and other popular plugins.
This plugin lets you search titles, post content, excerpt, categories, product SKUs, variations, authors, and posts with specific statuses (some of these features are available in the premium version). You also have the option to exclude specific content from search.
Ivory Search supports multilingual search. You get to create unlimited search forms and configure them separately. The plugin converts any search form to an Ajax one. More information is available here.