The ISU (International Skating Union) sets the rules for competitive ice skating including ice dancing. In competitions certain segments are required which make up for the overall results. The dances are skated on the hockey field and they have a set pattern. Normally you dance the dances with the same partner.

Until the end of the 2009/10 season compulsory dances, an original dance and the free dance were part of competions.
Compulsory Dances: All the dance teams performed the same standard steps. They are also danced by recreational dancers and are known as the Preliminaries or ISU-Dances (e.g. Dutch Waltz, Foxtrott,...)
Original Dance: The dancer had to dance to a designated rhythm and a specific theme. They were free to choose their own music and choreography.

At the start of the 2010/11 season the short dance was introduced, replacing the compulsory and original dances.
Short Dance: The dancers have to perform a required pattern, but for the second half of the dance they choose their own choreography with some required elements to a specified theme or rhythm.
Free Dance: It's usually the final part of the competition. Teams choose their own rhythms and program themes (music). Certain elements like step sequences, lifts, dance spins and multi-rotation turns are required though. Creativity is a vital element of the free dance.

This terminology doesn't exist in round dancing, though certain elements can relatively easily be integrated in the round dance.

sources: www.isu.org & https://en.wikipedia.org/wiki/Ice_dancing