I don't know much about the Mormons, so I can't say what prompted them to put together their FamilySearch website. But this is a wonderful tool for digging fast and deep into family history, especially for those with European roots.
FamilySearch is largely a pointer system and resource center. By typing in your ancestor's name along with other known information -- year of birth or death, country or region where they lived, parents or siblings -- you can begin to unearth family history information, and build a family tree.
Try both the basic search and the advanced search to get a feel for the site.
Your best bet is to simply play around at the site and explore their many resources. Enter a family name and see what pops up. Check out the Research Helps articles and guides. Take note of the free Personal Ancestral File (PAF) software that you can use to organize your family genealogical information.
Thank you Latter Day Saints. Thank you.