Faceted search is also called faceted navigation or faceted browsing.

This is an approach to web search that combines navigational search and direct search to allow users to navigate a multi-dimensional search space using search terms and then progressive filtering of the results to achieve improved resolution on the search.

The best way to understand this through example. Consider the site search on Research and Experts which is a very large database of industry and market reports.

1. Click on the report search tab
2. Type in a direct search phrase like 'aircraft industry in europe' and press search
3. Refine the search by selecting a particular region from the menu on the left (e.g. France)
4. Progressively refine the search by selecting other terms from the menu on the left until the desired result is found 

Each refinement on the initial direct search improves the relevance of the result.

The key to the success of faceted search is that it combines the best of direct search and navigational search. Direct search is used to provide an overall view of the search space for a given search phrase, these results then are refined using navigational search and the relevance is improved with each selection from a set of categorised options until the final highly relevant result set (the the users' requirements) is found.