The one word that characterises jazz is – improvisation. Jazz players improvise around a tune or theme, usually working with the “changes” (the chord/key changes) of the piece’s structure. Jazz improvisation can seem random, but it’s anything but. Top jazz players know their stuff – from classical scales to “modes” (other forms of scales) to harmony and, of course, complex rhythms.
Jazz is generally seen as having been born in the early 20th century, growing out of blues and developing into more complex forms. The great American songs – often from the musicals – were taken up by jazz players who also wrote their own pieces (originals), some of which have become jazz “standards” which all serious players must become familiar with.
To learn to play jazz, you need to master your instrument, develop your ear and be prepared to experiment and make lots of mistakes!
Over time, you will also need to learn how to read more and more complex music off the page.