Today’s hornpipe comes from the murky depths of the 18th century. Like many, it’s both a tune and a dance; the dance is a simple contra dance (first couple down the outside, up, down the centre, up, cast off, swing 6 hands quite round, right and left) noted in Elias Howe’s 1862 American Dancing Master and Ballroom Prompter. Hornpipes generally are also danced as solo fancy dances; as far as I know the authentically correct way to resolve any solo vs. group dance conflicts that come up is to dance as a group until someone has had enough beer to start dancing on the table.
The Library of Congress maintains three recordings of the Fisher’s Hornpipe, all by a guy called Henry Reed: in G, in D, and on the harmonica. Henry was evidently quite the authentic folk musician, where “authentic” is code for “not very good but has fun anyway”. For a better version, try this one from the 2nd South Carolina String Band.
There’s a fourth recording that’s also labeled Fisher’s Hornpipe, which is in fact a completely different tune better known as Soldier’s Joy.