So where better than to start at the beginning (of the audio recorded history that is). As the wikipedia entry pointed out the origins of jazz are strictly American - primarily from the deep south and the emergence of the dixie style. I won't endeavour to explore the African routes of the music going back into the 19th century, although surely a topic for a different time. However, Art Blakey would surely disagree!
The origins of jazz are clearly shown in the movement that came out of New Orleans at the very beginning of the 20th century. The primary instrumentation - cornet, trumpet, trombone, piano, string bass and drums. The essence of "Dixieland" jazz is the fusion of these instruments around the main melody. The trumpet usually takes the lead and the other instruments move around it. The influence on later music is clear to see - just listen to any Fats Domino record.
So what's acknowledged as the first recorded jazz song is Livery Stable Blues from the Original Dixieland Jass Band.
I always think that it is very eerie to hear such very early recordings. They contain an almost ghost like quality. To consider that when this was being recorded in New York in 1917, WW1 was being waged across the Channel.
Other notable songs to come out of this era were Tiger Rag, Margie and Royal Garden Blues.