Gimmicky baseball pitch, which essentially is an overhand version of slow pitch softball pitch.
The eephus is slowly lobbed up in the air, with a high arc, and drops down as it reaches the batter and catcher. It's so slow and high, it's sometimes called a "floater."
The pitch was first used by Rip Sewell, a pitcher for the Pittsburgh Pirates in the late 1930s and 1940s. One of his teammates, Maurice Van Robays gave the pitch its name, explaining "An eephus ain't nothing. And that's what that pitch is...nothing." (Some websites credit Sewell with the quote or a variation of it instead of Van Robays. Regardless, Sewell used the name thereafter).
Sewell had success using the pitch. Major league hitters aren't used to the slow speed or high arc of the pitch. As such, they're off-balance. However, if they do time it right, chances are it'll go pretty far. Ted Williams teed off on a Sewell eephus for a homer in the 1946 All Star Game.
Various pitchers over the years have used the quirky eephus once in a while, but none with the frequency of Sewell. It's generally a gimmick pitch, used for amusement of the player and the fans.
Journeyman pitcher Orlando "El Duque" Hernandez has taken to throwing the eephus once in a while. On August 26, 2002, he threw a few to Alex Rodriguez, who hit one out for a homer.
Rodriguez explained: "You can expect a lot of different arm angles and a lot of different pitches from him, but I never thought I'd get a 48 mph curveball. It looked like a slow pitch, softball really."
Sources:
http://sports.espn.go.com/mlb/recap?gameId=220826110
http://www.sportsjones.com/sj/489.shtml
http://www.pubdim.net/baseballlibrary/features/experts/05_02_01.stm
http://www.thebaseballpage.com/features/eephus.htm