Very simply, we can think of chords as more than one note sounding at the same time. We all know what a melody or tune is, and we play them on the saxophone. And of course the melody generally only has one note at a time. However in most jazz, pop, blues, rock etc. there are other notes in the backing (whether instrumental or vocal) that form chords or harmony. Often (but not always) the chords may change as the melody progresses. When this happens we call it the chord changes (noun), the  chord  sequence or the chord progression. On sheet music you will see chords written as chord symbols under the melody (e.g. C, D7, Am7 more on that later)
If you are not intending to improvise and just want to play or sing melodies, you never really need to know about chords, although knowledge of harmonic theory is always useful. But when you improvise, or maybe just want to embellish the melody, then it is important to know the harmony going on underneath in order that the (single) notes you play fit nicely with the mulitple notes (chords or harmony) that are being played or sung at the same time.
With a typical jazz or blues song, the melody is played over the chord progression. Then the chord progression is repeated while different players improvise over that progression, and then return to the melody at the end. The opening melody is referred to as the head, the repeats of the progression for improvising are called choruses, and finally the head is played again to finish off.