The story involves Jack and the Devil, but it has nothing to do with a fiddle.
The Legend of "Stingy Jack" starts in a tavern as any good story should. Stingy Jack invited the Devil to have a drink with him at the bar. He then convinced the Devil to turn into a coin in order to pay for their drinks. Once the Devil did so, Jack changed his mind and placed the coin in his pocket next to a silver coin with a cross on it. This didn't allow the Devil to change back into his original form.
Jack freed the Devil on the condition that the Devil wouldn't bother him for one year, and, if Jack should die, that the Devil would not claim his soul. Once the year had lapsed, the Devil showed himself to Jack again. Jack convinced the Devil to climb a tree to pick a piece of fruit for him. Once the Devil was up in the tree, Jack carved a cross in the tree's bark. This forced the Devil to agree to not bother Jack for ten more years and to not collect his soul for that duration should he die.
Soon after, Jack indeed did die, and God did not want the likes of Jack in Heaven, and the Devil couldn't collect his soul to take him down to hell. The Devil gave him a burning coal and sent him on his way through the dark night. Jack put the coal into a carved out turnip and has been roaming the earth ever since. The Irish began to refer this ghost as "Jack of the Lantern", and eventually, Jack O'Lantern.
In Ireland and Scotland, residents began to carve scary faces into turnips and potatoes to keep evil spirits like Jack away. When the Irish and Scottish immigrated to the United States, they brought their tradition with them, but they found a better alternative to carve: the pumpkin. The pumpkin is native to North America and serves well as the chosen produce to carve scary faces into for the night when we are closest to the dead.
And yes, the Headless Horseman of Sleepy Hollow was greatly influenced by the Legend of Stingy Jack.