The wall in the left field of Boston's Fenway Park. The wall is about 25 feet tall and robs many a batter of a homerun, hence the monster. The walls are primarily green at Fenway, hence the "green".
It is also the nickname of the lame new mascot for the Red Sox. Don't get me wrong, they are my favorite baseball team, but Wally the Green Monster mascot guy in a suit sucks.
Ex 1: "Trot Nixon blasted a 2-out opposite field dinger over the GREEN MONSTER to put the sox on top in the bottom of the eighth..."
Ex 2: "Kids these days are getting screwed out of cool mascots. I guess Wally the Green Monster is cooler than the MSU Sparty..."