A recumbent bicycle, or recumbent/bent, is a human powered vehicle which may have either two wheels or three. Sometimes the latter will be referred to as a "three wheeled recumbent bicycle".
Some allow the individual to pedal with their hand. These are called handcycles (or racing wheelchairs) and are typically more expensive than conventional recumbents.
As J. Totale notes, there is the question of visibility due to the height disparity between recumbents and automobiles. The solution to this would be to avoid motorways, but not all communities can boast of a proper bike path or track. This is a problem that has been mediated somewhat with the use of flags for those intrepid enough to ride with car traffic.
The idea that recumbents are not popular due to the stipulations of professional racers is a bit silly to me. More and more people today are using recumbents simply to make their heart race, not to haul ass. But yes, the recumbent will get up to a nice clip, if you let it. So a helmet is a good idea, as is being certain that your terrain is level and relatively free from debris.
Much like a decent bicycle, a recumbent will cost anywhere from 250 USD on up to a few grand, depending on how fancy a ride it is. Think of it as an investment for your health.
Attaching a boombox to a recumbent is a good way to make your ride more exciting.
People who use recumbents are typically older but not always—children's models are available.
A good resource for learning more about recumbents is the Recumbent Journal. Some builders are able to make special frames while others, such as Steve Schwartz of Bay Trail Trikes, can customize existing models to order.