Round dancing is partner dancing, similar to ballroom, except with the steps choreographed and cued, so there is *much* less emphasis on lead and follow. Popular in US Western square dance club settings.
Round dance may be described as a choreographed ballroom dance. Round dance clubs have a "cuer" who uses a microphone to tell dancers what to do, much like a square dance caller, but instead of a square dance, they are doing a pre-choreographed ballroom dance routine.
Round dancing uses ballroom steps, but the leader (called the cuer) calls out the steps to be done in a manner similar to the way that a square dance caller tells the square dancers what to do. Thus, all of the couples are going around the floor doing the same steps at the same time. A choreographer has chosen the steps so that they fit the piece of music being played.
Thus, while line dances and sequence dances have an unvarying pattern no matter what the music actually does, a round dance choreographer can account for the fact that some arrangements of a piece of music may have an extra measure or two sandwiched in between the longer 8 or 16 measure phrases.
Round dancing is closely associated with square dance, since at many square dances, the will alternate 2 squares with 2 rounds, but there are events where only rounds are done. At the easiest level only waltz and two-step (different from Texas Two-step) are done, but at the higher levels foxtrot, jive, rumba, cha cha, bolero, paso doble, tango, and other rhythms are added. The International Round Dance Teachers Association (usually called Roundalab) is the main group for round dance teachers and there are many other groups for both teachers and dancers.
Or, if you like a very short answer: Round dance is FUN!