Ballroom dance is a broad term that captures several styles of partnership dances. Examples of ballroom dances include everything from waltz and foxtrot, to rumba and bolero, to swing dances, among others. Ballroom dancing can be social, competitive, or performative.
Dancesport is the word for competitive ballroom dancing, as a regulated sport. Dancesport is generally divided into International (Latin and Standard) and American (Smooth and Rhythm) styles of dance. There are other ballroom dances outside these main styles, including Salsa, Hustle, West Coast Swing, Merengue, Lindy Hop, Polka, Country 2-Step, Nightclub 2-Step, and Argentine Tango, among others.
A more in-depth description of dancesport, ballroom dance, and styles of dances is discussed in "Ballroom Flow" from Kate Bratt's blog Riot and Frolic.