What is the difference in alcohol content and alcohol proof?

The proof number is twice the percentage of the alcohol content measured by volume, Therefore '80 proof' is 40% alcohol by volume.
The measure of the percentage of alcohol in drinks is widely believed to have started in the 18th century, when alcohol and gunpowder would be mixed together, if the combination lit when a flame was applied, it would be considered to have 'proved' it was a strong alcohol. (This is where the term 'firewater' is believed to have come from.) Originally that was considered 100 proof and was actually 50% alcohol, but more sophisticated measures since then have established that this alcohol and gunpowder mix would ignite if there was at least 57% alcohol, not 50%.