1. Duck Duck Go
At first, DuckDuckGo.com looks like Google. But there are many subtleties that make this spartan search engine different. DuckDuckGo has some slick features, like 'zero-click' information (all your answers are found on the first results page). DuckDuckgo offers disambiguation prompts (helps to clarify what question you are really asking). And the ad spam is much less than Google. Give DuckDuckGo.com a try... you might really like this clean and simple search engine.
2. Ask (aka 'Ask Jeeves')
The Ask/AJ/Ask Jeeves search engine is a longtime name in the World Wide Web. The super-clean interface rivals the other major search engines, and the search options are as good as Google or Bing or DuckDuckGo. The results groupings are what really make Ask.com stand out. The presentation is arguably cleaner and easier to read than Google or Yahoo! or Bing, and the results groups seem to be more relevant. Decide for yourself if you agree... give Ask.com a whirl, and compare it to the other search engines you like.
3. The Internet Archive
The Internet Archive is a favorite destination for longtime Web lovers. The Archive has been taking snapshots of the entire World Wide Web for years now, allowing you and me to travel back in time to see what a web page looked like in 1999, or what the news was like around Hurricane Katrina in 2005. You won't visit the Archive daily, like you would Google or Yahoo or Bing, but when you do have need to travel back in time, use this search site.
4. Yippy (formerly 'Clusty')
Yippy is a Deep Web engine that searches other search engines for you. Unlike the regular Web, which is indexed by robot spider programs, Deep Web pages are usually harder to locate by conventional search. That's where Yippy becomes very useful. If you are searching for obscure hobby interest blogs, obscure government information, tough-to-find obscure news, academic research and otherwise-obscure content, then Yippy is your tool.
5. Yahoo!
Yahoo! is several things: it is a search engine, a news aggregator, a shopping center, an emailbox, a travel directory, a horoscope and games center, and more. This 'web portal' breadth of choice makes this a very helpful site for Internet beginners. Searching the Web should also be about discovery and exploration, and Yahoo! delivers that in wholesale quantities.
6. Bing
Bing is Microsoft's attempt at unseating Google. Bing used to be MSN search until it was updated in summer of 2009. Touted as a 'decision engine', Bing tries to support your researching by offering suggestions in the leftmost column, while also giving you various search options across the top of the screen. Things like 'wiki' suggestions, 'visual search', and 'related searches' might be very useful to you. Bing is not dethroning Google in the near future, no. But Bing is definitely worth trying.
7. Mahalo
Mahalo is the one 'human-powered' search site in this list, employing a committee of editors to manually sift and vet thousands of pieces of content. This means that you'll get fewer Mahalo hit results than you will get at Bing or Google. But it also means that most Mahalo results have a higher quality of content and relevance (as best as human editors can judge).
Mahalo also offers regular web searching in addition to asking questions. Depending on which of the two search boxes you use at Mahalo, you will either get direct content topic hits or suggested answers to your question.
Try Mahalo. You might like it enough to even become a editor there.
8. Dogpile
Years ago, Dogpile was the fast and efficient choice before Google. Things changed, Dogpile faded into obscurity, and Google became king. But today, Dogpile is coming back, with a growing index and a clean and quick presentation that is testimony to its halcyon days. If you want to try a search tool with pleasant presentation and helpful crosslink results, definitely try Dogpile.
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9. Webopedia
Webopedia is one of the most useful websites on the World Wide Web. Webopedia is an encyclopedic resource dedicated to searching techno terminology and computer definitions. Teach yourself what 'domain name system' is, or teach yourself what 'DDRAM' means on your computer. Webopedia is absolutely a perfect resource for non-technical people to make more sense of the computers around them.
10. Google
Google is the undisputed king of 'spartan searching'. While it doesn't offer all the shopping center features of Yahoo!, Google is fast, relevant, and the largest single catalogue of Web pages available today. Make sure you try the Google 'images', 'maps' and 'news' features... they are outstanding services for locating photos, geographic directions, and news headlines.