Of the ten planets of our solar system Vulcan is (or was...) the nearest to the sun. Okay, so maybe there are only nine planets, but Vulcan should have been there. It was discovered by the same tried and true method that was used to find Neptune: Mathematics.
The facts: Uranus doesn't follow the path it should according to Newtonian Physics. Nor does Mercury.
The solution: Both of these facts could be explained by simply adding on two more planets; Neptune and Vulcan. The gravity of these 'new' planets would explain the abnormalities in the orbits of Uranus and Mercury.
After a time, it was discovered that Vulcan doesn't exist quite as much as it should. Mercury's orbit is actually a 'straight path' -- thru bent space. The gravity of the sun is so strong that it warps the space around it, making Mercury seem to wobble in its orbit. Newton had no idea... but Einstein's theory of gravity has cleared things up. So Vulcan has been made redundant, and has been discarded as a mathematical construct.
Occasionally a 'ghost planet' is seen in close orbit around the sun, but this is apparently due more to humans and their imaginations than any actual rocks in space. The last sighting of Vulcan was in the early 70s. Current belief is that Vulcan really and truly doesn't exist.
However, there is another planet Vulcan. In the Star Trek universe, it is the planet on which the Vulcan Race lives. It was discovered/invented by Gene Roddenberry, and it resides in a completely different star system from Earth and Sol. Both the planet and the race of Vulcan are famous primarily because of Spock, and important character in the Star Trek universe.
Many people would hold that this planet Vulcan is also imaginary.
But! Scientists have recently found a Jupiter-sized planet orbiting Epsilon Eridani. This is the star that some Star Trek fans claim that the Vulcan's home planet orbits around. It is most likely not an M class planet, and therefor not the *ahem* real Vulcan. It is also worth noting that Gene Roddenberry himself believes that the Vulcan race actually comes from 40 Eridani, not Epsilon Eridani. But, this Vulcan does really exist, so we'll cut it some slack.
There is also a town in Alberta, Canada called Vulcan (Vulcan is also the name of the county). It is most notable for its small (9.4488 Meters long) Constitution Class Starship (Vulcan Starship FX6-1995-A). Despite the starship, it was named after the Greek god, not the Star Trek character/race/planet.
Thanks to locke baron for catching a factual error, and Space.Com for providing some important facts in the first place.