The concept of "Swing" is elusive to most beginners. It can be difficult to understand at first, because it requires comprehension of movement, and most newcomers tend to see dancing as a series of positions. But I will do my best to clarify...
Sway is the inclination of the body to one side or the other. Pitch is the inclination of the body forward or backward. Whereas pitch and sway can be considered either an action *or* the resulting position, swing can only be an action. There is no "position of swing" that you could identify in a photograph the way you could pitch or sway.
Also, pitch and sway have very specific ranges: They are both a tilting of the body to one direction. Pitch is strictly forward or back, while sway is strictly to one side or the other. Swing, on the other hand, can happen in a number of different ranges, forward to back, side to side, diagonally, or even rotationally. A swinging action can therefore result in pitch, sway, turn, or any combination thereof (as well as none of the above) depending on the range it's danced in.
If I were to define the word swing in ballroom dancing terms, I would describe it as "the free release of movement through an arc". So for a movement of the body (or a part of the body) to be described as "swinging", by this definition it would need to have two qualities: (1) It would need to appear to move in the range of an arc, and (2) It would need to appear to move freely, as if "let go". This almost always involves a certain degree of acceleration, and a subsequent natural tapering-off of speed. If a movement does not possess both of these qualities, it would not likely be perceived or described as "swinging" in ballroom terms.
Swing is often divided into one of three categories: (1) Pendular, like the swinging of a golf club. (2) Metronomic, like the swinging of an axe. (3) Rotary, like the swinging of a baseball bat.
When we're discussing body swing, the first two -- pendular and metronomic -- can be taken side to side, which results in sway, or forward and back, which results in pitch. You could also swing through a diagonal, which would result in both pitch and sway. The last one -- rotary swing -- simply results in a turning of the body.
What makes even the most basic actions in smooth ballroom so interesting is that they involve a complex combination of many of these ranges together, to varying degrees at different points. For example, a basic Waltz Natural Turn is a combination of rotary and pendular swing. Since it begins with a forward step, the pendular motion starts to create a slight pitch of the body toward the end of the step, but this transforms into sway as the body turns, causing the forward movement to become sideways.
If you're more confused than when you started, not to worry... You can print this out and take it to a teacher who will demonstrate these ideas to make them clearer. They're much easier to see than to read about.
Regards,
Jonathan