Syncopation
Perhaps more so than the melody or harmony, it is the rhythm of a performance that identifies it as jazz for many people. Jazz developed in part from ragtime piano music, and the very name "ragtime" is a contraction of "ragged time", which refers to the use of syncopated rhythms that are common in both ragtime and jazz.
In the most general sense, syncopation has to do with the creation of rhythmic surprise. The most typical way this is done is through the placement of accents. In most other styles of music in the Western world, accents generally fall on the beats:

In jazz, however, accents tend to fall between the beats. If you divide each beat into two parts, the accents in jazz often fall on the second half of each beat. Furthermore, they appear irregularly; you do not tend to see the second half of every beat accented equally:

Note the accents in this example occur within the melodic line. One of the characteristics of bebop in particular is the use of short, accented offbeats at the ends of phrases:

Another aspect of syncopation has to do with which beats are emphasized. In other types of music, when there are four beats per measure (as is most common), beats one and three are usually stronger than two and four:

In jazz, at least since the Swing Era, beats two and four - the "offbeats" - often feel stronger than one and three:

All four beats are still played, but beats two and four are accented slightly. This is sometimes referred to as a four-beat rhythm. When two and four are accented especially heavily, it is sometimes called a "backbeat" rhythm:

The early jazz musicians often strongly accented beats one and three, however, with bassists and drummers playing almost entirely on these two beats, and pianists or guitarists supplying chords on the offbeats:

This is referred to as a "two-beat" rhythm. Modern jazz musicians will