A simple description of syncopation is the creation of rhythmic surprise. This can often be done by accenting certain notes or by unevenly timing the beats. Both jazz and ragtime make use of syncopation; the name of ragtime music itself comes from its uneven or "ragged timing."
When music is played as it is written on the page, it is said to played straight, or unswung. In other words, if what is written on the page of music are 8th notes, and you play those notes as perfectly timed 8th notes -- each having the same duration of time -- then you are playing the music straight.
When it comes to swing, music and syncopation, swing refers to the creation of the feel of a forward momentum, a forward momentum that seems to want to drive someone to dance and sway to the music. Often this is done by dividing each beat into a pair of 8th notes, but playing the first slightly longer than an 8th and then playing the second slightly shorter than an 8th, or vice versa. This is sometimes described as the first note "stealing" some of the second note's time so that they are of unequal duration. This is often called "swing time."
Another way the music is "swung" is by playing the just a fraction of a beat ahead of the actual beat giving the music as a whole a feel of forward momentum. On the other hand, playing just behind or slightly after the beat can give the music a relaxed feel.
Music that has swing is often exciting or at least makes you want to dance. Music that does not have swing often feels rigid, boring and lifeless.