What is Round Dancing?
Round dancing is partner dancing, similar to ballroom, except with the steps choreographed and cued.
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.
In many cases the dances have been choreographed to very carefully fit the music and take advantage of interesting spots. You learn the names of all the moves you're doing and you can usually transfer the moves (and good sequences) to the ballroom dance floor. And lessons are much cheaper.
Another answer: 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.
At many square dances, they will alternate 2 squares with 2 rounds, but there are events where only Round Dancing is done. Various rhythms such as waltz and two-step (different from Texas Two-step) are done and also 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 Dancing is FUN!