It's a Google world, at least that's what most pundits would have you believe. It's certainly true that by most standard metrics, Google do have the largest share of web search market, with some estimates pegging it at over 75 percent. While that figure is large and growing, there are other search engines that rival Google for speed and accuracy.
The second most popular search engine is Yahoo. Yahoo use a different methodology to Google in assembling their results, relying on a 'web crawl process'. 99 percent of web pages that Yahoo index are included free, however some results might also include links to sites that participate in the Yahoo's content acquisition program that lets content providers pay Yahoo to 'index' their content.
Another serious competitor in web search is Microsoft. Microsoft's search product is called Live, and it's located at live.com. Live Search offers regular text based searching (alaGoogle and Yahoo), but Microsoft are also making an effort to include other media, especially video, in their search results. A new 'inline preview' feature enables you to view videos without leaving the search results.
It's also worth remembering that there are also search engines for different kinds of media. If you're searching for an image for example, a popular image search engine isPicsearch.com, which searches two billion images.
If you can't decide which search engine you prefer you can always find a search aggregator (sometimes called a metasearch), where all search engines are available from one search box. A good example of this concept is Dogpile, where you can search Google, Yahoo, Live Search, Ask, About and Looksmart in one interface. Dogpile does the work for you and sorts out the results based on relevance.
Whichever search engine you decide to use, you should be comfortable that between Google and the rest you'll likely to find the results you are looking for.