A barium swallow is a way to view the upper digestive tract, namely the esophagus. The patient swallows a solution of barium sulfate, and the swallowing process is viewed through a fluoroscope. It's useful in detecting ulcerations, cancer, and obstructions.
Also used to "see" the upper digestive tract, the barium meal is a procedure used to see what is going on in the esophagus, stomach, and duodenum. The patient has his stomach expanded, then swallows the barium solution. After it has sufficiently coated everything, an x-ray is taken. More intensive than the "swallow," the "meal" spotlights growths, lumps, abnormalities, etc.
The barium enema, on the other hand, is a procedure used to "see" the lower digestive tract. Anyone who's ever had an enema knows that a tube is inserted in their rectum and a solution forced up into the lower bowels (colon). The barium coats the colon, enabling a clear view of anything that's wrong with that area of the body.