I once attended a workshop where Joe O'Donovan, the old time step dancer and teacher from County Cork in Ireland (I believe), taught an old time version of this set dance. He indicated that the title referred to three sea captains under Admiral Nelson (possibly at the Battle of the Nile, I can't recall for sure.) There was at least one place in the dance where the dancer struck the floor with her/his toe (wearing hard dance shoes with fiberglass tips), indicating the firing of cannon.

Some of the earlier postings, citing older sources for the tune, may call this explanation into question.

In Irish dance music a set dance is a tune of non-standard structure, standard structure being 1) each tune part consists of 8 bars of music, and 2) all parts are in the same time signature. In the case of The Three Sea Captains (at least as it is played in Irish music) the "A" part of the tune is 8 bars long (standard), but the "B" part is 20 bars long (non-standard.) Perhaps this 20 bar "B" part consists of several 8 bar parts, or fractions thereof, that have been melded together - I cannot say.

Alternatively, "set dance" may refer to a prescribed series of steps that have been "set" to such a non-standard tune. Since the tune is non-standard, the dancer cannot use her/his garden variety steps, and so must have special steps prepared.

I apologize if I have merely repeated information contained in Dicho's web reference above.