A syncopation or syncopated rhythm is any rhythm that puts an emphasis on a beat, or a subdivision of a beat, that is not usually emphasized. One of the most obvious features of Western music, to be heard in most everything from Bach to blues, is a strong, steady beat that can easily be grouped evenly into measures. (In other words, each measure has the same number of beats, and you can hear the measures in the music because the first beat of the measure is the strongest.) This makes it easy for you to dance or clap your hands to the music. But music that follows the same rhythmic pattern all the time can get pretty boring. Syncopation is one way to liven things up. The music can suddenly emphasize the weaker beats of the measure, or it can even emphasize notes that are not on the beat at all.
Syncopations can happen anywhere: in the melody, the bass line, the rhythm section, the chordal accompaniment. Any spot in the rhythm that is normally weak (a weak beat, an upbeat, a sixteenth of a beat, a part of a triplet) can be given emphasis by a syncopation. It can suddenly be made important by a long or high note in the melody, a change in direction of the melody, a chord change, or a written accent. Depending on the tempo of the music and the type of syncopation, a syncopated rhythm can make the music sound jaunty, jazzy, unsteady, surprising, uncertain, exciting, or just more interesting.