It started off as slang term for communion hosts in the 16th century (blasphemous). Later it was a way of signing a petition in a circle so that the leader or originator of the petition (usually the first to sign) could not be deduced by the signatures (18th century, used by royal navy grievance petitions) Now it is a form of competing where everyone plays, versus an elimination-style tournament.

From http://www.word-detective.com/back-m.html#robin Q:Dear Mr. Morris: I wonder if you can be of help. I work for Taiwan's Trade Office and we are planning to have a "round robin" party at each of our houses. Upon hearing the phrase "round robin" our Director was very interested to hear why we use such a phrase and its origins. As far as I can tell from the dictionary, this is a term used to describe, besides a game-playing tournament, a petition of signatures written in a circle so no one can be blamed as the ring-leader. Any additional help and clarification of this phrase's origin would be very interesting. A:It's nice to see that the tradition of "round robin" parties is alive and well, and in the Taiwan Trade Office, no less. I still remember the last round robin party I went to, on the Fourth of July weekend in 1970. I was going to say that I "vividly remember" the party, until I realized that just about the only thing I remember of it is that it lasted for three days. Ah, youth. In any case, you're on the right trail as to the derivation of "round robin." The phrase actually has nothing whatever to do with a bird, robin or any other kind. "Robin" in this phrase is a corruption of the French "ruban," meaning "ribbon," and the origin of the phrase is about as far from a party as one can get. It seems that in 17th and 18th century France, there was a good deal for the average peasant to complain about, but complaining to the King in particular was not a terribly good idea. The monarch's usual reaction to a petition from his subjects was to seize the first two or three signers and have them beheaded. Not wishing to lose their heads, but bent nonetheless on petitioning for justice, clever peasants came up with the expedient of signing their names on the petition in a circle, like a ribbon. That way, no one's name came "first," and, assuming that there were hundreds of signatures on the petition, it was impractical for the King to punish all the signers. A similar method was adopted by disgruntled sailors in the 18th century British Royal Navy, another institution not known for welcoming criticism. Sailors often signed their names to a petition like the spokes of a wheel, so that no one of them could be considered the leader of a mutiny and hanged. Today we use "round robin" to describe any event, most often a sporting event of some kind, where everyone takes a turn.