This is a baseball term that refers to the pitcher who comes in relief in the last or second-to-last inning to close out the game. A team that leads, or trails very closely, will bring in a pitcher capable of getting strikeouts at the end of a game. The end of a game is often the most stressful time and requires incredible accuracy, stamina and poise on the part of the pitcher. The closer plays an important role in the pitching rotation of any team, as he or she may be called upon to pitch multiple days in a row. Managers must closely monitor the number of pitches a closer throws, as these pitchers typically throw harder and faster than other pitchers, which wears out their arms faster.
Mariano Rivera of the NY Yankees is considered the most famous-and likely the best-closing pitcher of the modern age of Major League Baseball. He has won multiple accolades and is a sure bet for the Hall of Fame when the time comes. His consistency and ability to stay effective for so long has changed the perception of closing pitchers. Once considered a position for pitchers who couldn't start, the closer has morphed into a specialized position that requires a special skill set.
In baseball slang, the term "closer" also refers to a hitter who makes a lot of walk-off hits (i.e. hits that win games).