In professional wrestling, the term over has two definitions. It can either refer to a performer whom the fans care about (That wrestler is over as a heel; the fans hate his guts.) or the act of making someone look good, often by losing to them (Bret Hart put Shawn Michaels over by losing to him cleanly, but Michaels didn't return the favor.)
Wrestlers can be over as either babyfaces (good guys) or heels (bad guys). The term suggests that the fans are buying into what the wrestler is selling, meaning his character. "Over" has a denotation of the fans believing in the wrestler such that they are willing to boo and cheer his character as if his matches were not a work. Wrestlers who are over are generating reactions from the crowd, known as heat in wrestling parlance. Someone who is over as a heel will get booed, while someone who is over as a face will get cheered. The term likely originates from the days when promoters did not acknowledge that pro wrestling was scripted, and the wrestlers were thought to be tricking, or "putting one over" on the fans.
To "put over" something (a wrestler, promotion, match, etc.) means to make it look good or otherwise encourage the fans to care about it. This can be done in the ring, during a promo, or through many other kinds of media. One of the most common ways a wrestler can be put over is winning a match. For instance, "at the next pay-per-view, Kane is putting Chris Jericho over" means that Jericho will beat Kane. It's also possible to put someone over by taking spectacular bumps or selling a move. After Kevin Nash (as Diesel) defeated Bob Backlund for the WWF title, Nash said that Backlund "couldn't have put me over any stronger" because Backlund crawled back up the entrance ramp to sell Nash's powerbomb. A wrestler or announcer could put something or someone over in an interview or promo simply by complimenting them.
The phrase can also refer to something a wrestler does or says--a wrestler could be over with the fans, but not everything he says or does gets over. For instance, "The fans love Mick Foley, but that joke he told last night didn't get over; the crowd groaned instead of laughed."