There are dozens of genealogy projects currently active. They range from regional to global in scope. Some are meant to give a big picture glimpse of how people migrated from one region to another over millennia -- these are more anthropological than genealogical in nature. Others help people get in touch with fellow genealogists to solve mysteries and connect to family members who may be separated by geography and generations alike.
One of the largest projects is the Genographic Project spearheaded by National Geographic. You can participate in the Genographic Project by purchasing a test -- it costs around $100 -- and submitting your sample to the project. It's not meant to help you find out who your great-great-grandfather was. Instead, the project's aim is to map the migratory patterns of human history. It's about as big a picture as you can get.
The USGenWeb Project is a volunteer organization dedicated to helping citizens of the United States research their family backgrounds. The project has links to each state project. Within the state project site, you'll find links to resources that might help you find out more about your family. In many cases, the links will tell you where you need to go to see official documents that have your family's information on them. You'll still need to do some legwork to fill in the gaps, but the projects resources can give you a good place to start.
The WorldGenWeb Project has similar goals but on a global scale. It contains links to regional genealogical project Web sites. Volunteers can elect to oversee a particular region. It becomes that volunteer's duty to gather research resources and create forums for members to connect with one another and discuss family histories.
Many surnames have their own DNA projects. Most of these projects began as personal projects that grew over time and merged with other projects for the same surname.
Keep in mind that these projects are meant to help you in your search for information about your family. Don't expect them to present your complete family history through a single search query. You'll most likely need to do additional research and contact distant relatives to build out a full history.