A nickname is a name given to a Pokémon by its Original Trainer. Every time a player catches, hatches, or is given a new Pokémon in a main series game, he or she is able to nickname the Pokémon to a maximum of ten characters. In addition to the games, nicknamed Pokémon have also been shown in the Pokémon anime and several manga series. Nicknames serve as a means of personalizing one's Pokémon, and distinguishing them from other individuals of the same species.
Nicknaming Pokémon rarely has any effect on game-play, and is simply an element of customization that players are free to use or ignore. The only games in which nicknamed Pokémon are treated differently are Pokémon Stadium and Pokémon Stadium 2. In these games, when nicknamed Pokémon transferred from a main series game they have a chance of being alternately colored. This coloration is not the same as being a Shiny Pokémon. This feature has not been included in any later games.
Sometimes, in Pokémon Ruby, Sapphire, and Emerald, when the player gives a previously caught Pokémon a nickname at the Name Rater's house, the Hoenn TV network will report the nickname chosen. The host will always commend the player on his or her choice of name, even if the player decides to leave the Pokémon's name as it was. When records have been mixed with another save file, the television network may report on the other Trainer's choice of nicknames.
With Generation IV's Global Trade Station, some unfortunate naming issues may arise. The main issues are receiving Pokémon with nicknames in a foreign language and receiving Pokémon whose nicknames are threatening, inappropriate, racist, vulgar or pointless (for example, a name consisting of random characters). As the chances of matching both the Trainer ID and Secret ID are near impossible, these nicknames cannot be changed.
Pokémon with nicknames deemed "inappropriate" may show up in Battle Videos as Pokémon without a nickname: a Staraptor named inappropriately would have its nickname reverted to "STARAPTOR". It is unknown if this censoring is automatic or done on a case-by-case basis by Nintendo employees.
Yeah, so nothing much. If it's inappropriate, it won't show up in Wi-Fi, and it's just to personalize your Pokemon.