Vulcan was a theoreticalplanet. In the 19th century, the planet was thought to orbit close to the sun. Its gravitational pull would explain changes seen between the perihelion advance of Mercury from that predicted by Classical Mechanics. The theory that Vulcan exists come from the earlier prediction that an outer planet (now Neptune) was causing similar changes in the orbit of Uranus from the orbit predicted by classical theory.
None of the searches for Vulcan found a planet. This caused some people to not agree with Newtonian theory. It led to the development of Special Relativity and General Relativity. Albert Einstein's modified theory of gravity showed why the orbit of Mercury changes without the existence of Vulcan. This disproved the theories of Vulcan.