Braising involves cooking meat in a closed container with some liquid in it. The steam produced keeps the temperature relatively low compared to air heat, and this allows the inside of the meat to cook and tenderize without the outside of the meat overcooking and burning.
Braising is best done with cheaper cuts of meat that have fat in them to keep them moist, and muscle collagen in them that will gelatinize and make the meat juicy. Some recipes will also have you braise tender cuts of meat, but for much shorter times as they cannot stand up to the longer cooking times.
It is often advisable to sear the meat first, to develop the flavour on the surface of the meat (don't forget to swish some wine or stock in the braising pan to recover the tasty browning bits left in the pan.) Often, you will flour smaller bits of meat first (such as stewing beef) before browning. Brown the meat slowly, so that the surface can carmelize evenly to develop flavour.
The liquid you use to braise in can be water, wine, beer, stock or juice.
Just remember the catch-phrase: lo-n-slow. The liquid should not be bubbling with heat, but rather kept at a slow simmer which just causes the surface to tremble. The pot or dish in which you are doing the braising should be covered to keep the liquid in.
Many braised dishes taste even better the next day.
Literature & Lore
Harold McGee, in his excellent "On Food & Cooking: The Science and Lore of the Kitchen" (New York: Simon & Schuster, 1997, page 120 -- Braising) points out, in fact, that cooking meat in water in a covered vessel will in fact help to make meat drier. The hotter meat gets, the more water it exudes, and steam is a more effective conductor of heat than air is.