What is a private search engine?
Private search engines search for your given query without associating your search request with you or your browsing history.
Generally, private search engines don't try to associate the searcher with what is being searched, unlike "traditional" search engines. This enables users to maintain privacy while searching, limiting the inclusion of their search history into advertising and tracking profiles; it also limits the surveillance capabilities of public (and private) organizations.
Search vs Metasearch
Most private search engines are actually metasearch engines - meaning they pull data from other search engines such as Bing/Yahoo and Google. Typically, they strip identifying information from the query before passing it to Bing or Google.
Often, private search engine services will pull from many other search engines - outside of Bing or Google - from across the web, which could mean dozens or even hundreds of different sources. Many private search engines also run and manage their own web crawlers, which index pages of websites, for inclusion into search results.