Cosplay is a Portmanteau of the words "costume play". It refers to the fan hobby of making costumes of a concept or person and wearing them at fan gatherings, though is commonly associated with fictional characters from mediums such as anime, TV, movies, comics, video games, etc. At anime conventions cosplay is one of the most popular events.
And to some people, it is Serious Business. Don't even think about trying to get them to wear the Official Cosplay Gear. They will often wear a Sexy Whatever Outfit though.
If it's a character getting cosplayers in the same universe, the actual character might get told Your Costume Needs Work. This can even happen to celebrities in Real Life.
See also Cosplay Fan Art, Cosplay Otaku Girl (characters in works being cosplayers), Whole Costume Reference, LARP.
In episode 16 of Bodacious Space Pirates, pirate captain Marika has members of the Girl's Yacht Club help out on her ship, the Bentenmaru, while the regular crew is stuck in quarantine due to a virus they got in an earlier episode. When they raid a ship, the girls all show up in various cosplay costumes. Marika chuckles nervously throughout the entire "raid", but the guests on the ship don't seem to mind the costumes overall.
In Genshiken this is Ohno's hobby, while Tanaka's is costume designer for Cosplay. She often wears his creations for the club and to conventions. In Genshiken 2 they finally officially start dating, thus creating the dream Otaku couple.
In Genesis of Aquarion there is an episode called "Cosplay for the Soul" where Fudou has all the Element Users dress and act as each other to great (and hilarious) effect.
Honestly, this Troper was expecting Aquarion itself to cosplay as a different giant robot from another anime/manga during some point.
Persona 4: The Animation has Nanako doing a cosplay so convincing that she fools some kids into thinking she's the real Loveline (an in-universe, fictional, magical girl detective).
Spider Jerusalem at one point raids a religious convention while cosplaying as Jesus.
In one episode they all dress like The Flash for Penny's Halloween episode.
They go as the Justice League team for the comic book store's New Year's Eve costume party, with Penny as Wonder Woman and Penny then-boyfriend Zack as Superman.
The episode "The Barkersfield expedition" opens with Sheldon steaming his Data cosplay and Leonard having to borrow make up from his girlfriend for his. The A plot of the episode actually revolves around the guys getting stranded in the desert after pulling over on the way to ComicCon to do a full cosplay photo shoot. Leonard was dressed as Captain Picard, Raj as Worf and Howard as a Borg. They looked awesome.
In one episode, the girls decide to go to Disney land and they buy Disney Princess line-up with costume, make-up, hairdo... The only problem was that they all wanted to be Cinderella. Bernadette claimed it was her idea and that she was a driver, so she won. Penny was Aurora and Amy Snow White. Howard and Leonard really fancied their girlfriends in their outfits.
Litchi Faye-Ling of BlazBlue has a cosplay fetish that comes out when she's had too much sake, and whatever cute girl in arm's reach is usually subjected to it. So far, Noel and Tsubaki were victimized - Noel was coerced with playtime with Lao Jiu, while Tsubaki was a bit more cooperative... but just a bit. According to Arcade commentary, however, Makoto could be next...
Litchi herself is a pretty popular subject to be cosplayed as in real life.
One of the early Hijinks Ensue strips involves Josh's (real-life) cosplay as Kratos netting him Internet notoriety. His response is to create the most awesome costume ever created (this, sadly, did not also happen in real life).
The Closing Credits Sequence of Daria features the characters in various alternate costumes. Most of the time, the costumes are generic an inspecific, like a cheerleader or a business woman, but occasionally they make specific references like Brittany dressed as Edward Scissorhands.
When the cast of Futurama goes to Comic Con 3010, they take part in the costume contest, with some of the contestants dressed as characters from the very show they're on. Lrr tries to invade Earth during the contest, but is dismissed as just another guy in a costume. The winner, by the way, is Bender, who dresses as "every nerd's fantasy", Leela.