Search API with Apache Solr
The Search API Solr Search module page on drupal.org has a list of services that can host Apache Solr indices. Most inexpensive shared hosting providers don’t provide a Solr index, and your institutional IT staff may or may not be willing to provide one. Using an external service can give you access to Solr even if your core hosting doesn’t support it, but you may want to first try using Database Search and Search API to improve search on your site.
There’s information on drupal.org for installing Solr, configuring a Search API server, and customizing the Solr configuration.
Here’s a detailed YouTube tutorial on how to use Solr with Drupal.