Search is powered by a Core extension "Indexed Search". Configuration can be found in EXT:speciality file TypoScript/constants.txt and TypoScript/Plugin/tx_indexedsearch.ts. If you don't use the search feature it is advisable to deactivate the extension since it will consume resources by indexing the content while browsing the website.
As tools, once logged in the BE, the indexed content can be checked out in module "Web" > "Info" > "Indexed Search" (in the drop down menu on the top). There also is a BE module for stats in "Admin Tools" > "Indexing".
If you are looking for third party extensions, following suggestion can be made. For bigger website, Solr or equivalent can be considered. A bunch of extensions are available on the TER. Besides, Google Search plugins can be downloaded and installed.