A breaking ball (aka breaking pitch) is a pitch in which the pitcher snaps or breaks his wrist to give the ball spin and movement. This includes the curveball, slider, and slurve, but not the various kinds of fastball and change-up or trick pitches like the knuckleball.
The term "breaking ball" is used in a variety of ways. It is commonly used by broadcasters when they can't be sure whether a pitch was a curve or slider but know that it was one or the other. This is most common when a pitcher throws both a curve and slider, but also happens when the announcer isn't familiar with the pitcher's repertoire.
"Breaking ball" is also used as a way of lumping the curve and slider into a family. Some people see the slider and curve as being arbitrary points along a continuum rather than sharply distinguished pitches. Pitching coach Leo Mazzone, for instance, encourages his pitchers to develop a "quality breaking ball" without making a strong distinction between the different varieties of breaking pitch. This view is actually a very old one. In the 19th Century, it was common to call all breaking pitches curves. It's likely that some 19th Century pitchers threw pitches that were called curves back then but would be described as sliders today.