Syncopation in music is often misunderstood. It is, very simply, a deliberate disruption of the two- or three-beat stress pattern. Musicians most often create syncopation by stressing an off-beat, or a note that isn't on the beat.
In 4/4 time, the general stress pattern is that the first and the third beats are strong, and the second and fourth are weak. Another way to say this is that downbeats, or accented beats, such as those at the beginning or halfway through a measure, are strong, and upbeats, or unaccented beats, are generally weak.
Often, the fourth beat of the measure is accented instead of the third beat, which is normally accented, creating a different-sounding rhythm than you would normally have in 4/4 time music. Instead of ONE-two-THREE-four, the measure would be counted off as ONE-two-three-FOUR.
That's syncopation — the natural stress of the meter has been disrupted. The count ONE-two-(three)-FOUR is weird to your ear because you want to hear that nonexistent quarter note that would carry the downbeat in the middle of the measure.
If you do anything that disrupts the natural beat with either an accent or an upbeat with no subsequent downbeat being played, you have created syncopation. People often mistake syncopation as being comprised of cool, complex rhythms with lots of sixteenth notes and eighth notes, as often heard in jazz music, but that isn't necessarily true. Even if a piece of music contains an entire measure of eighth or sixteenth notes, it doesn't necessarily have syncopation. Every eighth note has a subsequent rhythmic resolution. In other words, the downbeats still occur in the measure where they're supposed to be, on the accented notes shown in the figure.
Consider this rhythm: ONE two three FOUR one TWO three FOUR. The natural stresses have been shifted over in both measures, resulting in a purposefully disjointed-sounding beat.
So does syncopation involve a carefully placed rest or an accented note? The answer is both. If your perspective of where the downbeat occurs is moved, a point of syncopation results because it's shifting where the strong and the weak accents are built.
Try counting out the beats while listening to the Rolling Stones' "Satisfaction," and you'll hear some great examples of syncopation.