Any type of competition
in which every player plays against every other player.
The term round robin may have originated in the 1600s, when french monarchs would often put the instigator of a petition to death. To prevent this, some bright soul invented petitions that were written on a circle of ribbon -- thus preventing any name from getting the first position on the petition.
British sailors later modified this by signing their names in a starburst, with all names starting at one central point, and radiating out like the spokes on a wheel. In the rare case that you hear of a round robin petition these days, this is what is being refered to.
A round robin poem is a poem in which people (usually young children) take turns adding on lines. These are sometimes called round robin rhymes, as the focus is most often to teach children about rhyming.
The ever popular writing exercise (or perhaps 'experiment' would be a better word) in which a piece of paper is passed around the group, with each person adding a line to the story, is often called round robin, also.