I apologize for reusing this Bela Fleck documentary again, but it's so full of great examples that I'm going to milk it for all it's worth.
While working, slaves would occupy their time by singing songs (most of us are familiar with Wade in the Water. These have become referenced to as works songs.
The most influential part of these work songs that has stuck through with jazz today is the call and response concept. In the video, when the leader of the song finishes her phrase, the rest of the singers response and sing a phrase back. This can also be seen in the work songs the slaves would sing. If you've ever listened to jazz and soloists will trade soloing every four measures or so, this derives from African call and response. (In this videocall and response is implemented in the horns responding to what the vocalist is doing).
This has become a vital element to jazz playing. When a soloist is going at it, all the other instruments playing at the time (drums, bass, guitar/piano, etc.) are expected to be listening and thus respond accordingly in order to respond and support the soloist.
Another important part of this history is the integration of slaves into the church. The whites approached preaching to the slaves with a "save the savage souls" mentality and so constructed churches for the slaves. From here, slaves could learn instruments and understand the basics of harmony and song writing. In the end the whole concept backfired because the music the slaves created was considered devil music by the whites.