A multi is an alternate to one's main character in a role playing game. Other names for this are alt and alt char.
The act of playing several characters at once is called multiing.
A multi may be used in order to round out one's party or help shoulder the load of treasure.
In the instance of tabletop gaming, it is pretty clear to all present who is using a multi and who is not.
Online is a different kettle of fish.
Some MMORPGs, such as Kingdom of Loathing, have special rules governing the use of multis, in order to curb instances suspected cheating.
Banning of multis outright is rarely done and not easily enforced. As such, there are a few stipulations which may be enacted to make the game "more fair" with regard to the use of multis. Among them are: limiting the number of characters an individual player may have, forbidding an individual player from having characters in multiple clans or guilds, allowing only one character being logged in at a time, forbidding a player's characters from interacting or sharing resources amongst themselves. This last limit may even extend to in-character dialogue.
In a tolerant playing environment, multiing can be a fun and therapeutic way to exercise the imagination. The determined individual can carry a multi-faceted conversation between their main character and their multis.
Such creative play would be encouraged in the case of a MUD, MUSH or muse.