What is the name of the first submarine?

That would be the U.S.S Holland. Actually that was the first U.S. commissioned naval submarine. The first actual submarine was the Turtle (invented by David Bushnell) used in the American revolution against the Brittish HMS Eagle. In 1620, a Dutchman named Cornelius Drebbel (who was working for England's James I) built a submarine that was designed by William Bourne, the Brittish mathematician. Only some general plans remain, but their authenticity is not questioned. This undersea boat predates the Holland and Turtle. A link is provided to the Wikipedia article on the history of submarines. Our friends there post knowledge for free.
The first real sub to use a propeller and in war time was the TURTLE. Built by David Bushnell in 1776, and piloted by Sgt. Lee of the Continental Army.