Well, that's easy. The best free people-finder site is Google, of course.
Depending on who, and what, you're looking for, and how common or unusual their name is.
Try three types of Google searches: the name, the name in quotes ("John Smith"), and the reverse name in quotes ("Smith John"). And remember that someone you know as Bob or Mattie may be listed as Robert or Mathilda in various places, so play around with that first name.
Search Google News as well, and while you're at it, search at NewspaperArchives.com (search is free, but there's a charge to retrieve articles). You'll be amazed at how many people get mentioned in newspapers for all sorts of reasons, from weddings to arrests!
Try your search at Pipl.com, one of the best web-scraping sites that shows a scary collection of personal information from Facebook, Linked-In and elsewhere on the Internet. Don't forget a search in the phonebook, too...I usually use SuperPages.com for both people and business searches.
Look in specialty databases. CriminalSearches.com has pretty deep arrest records. Jigsaw.com for business listings (free search, but pay for full records), and MissingMoney.com for (believe it or not) people with abandoned bank accounts (search for your own name...you never know!).
Check the Social Security Death Index if you think the person you're seeking may have gone to the great beyond.
Intelius is the best of the commercial people-finders. They have the largest dataset and, in my experience, produce the most reliable results. You can do a preliminay search for free and get some pretty useful information, but you have to pay for a full record. Be forewarned...I've heard complaints about their customer service, so be careful with the site.
Need more information? Newspaper archives are one of the best online research tools.
If you're searching for information on people, try Intelius.