Vulcan is the hypothetical closest planet to the sun. It has been "discovered" several times in the last 150 years, but each time it has been disproven.
Vulcan was first speculated in 1859 when when an an amatuer astronomer spotted the "planet" orbiting the sun. The astronomer (Lescarbault), reported this "planet" to Urbain Le Verrier (a French mathematician and astronomer), who named the planet "Vulcan". This first incarnation of Vulcan was said to be orbiting at an inclination of 5.3 to 7.3 degrees with a longitude of 183 degrees, and was said to have taken 270 minutes to traverse the visible disc of the sun. Le Verrier took this data and computed that Vulcan was a planet having 1/17th of the mass of Mercury, with and orbital period of 19 days 7 hours at a mean distance of .1427 au from the sun.
In early 1860 Le Verrier and J.C. Adams postulated that the orbital variations of Mercury could be explained by an intra-mercurial planet or asteroid belt. He had already successfully predicted Neptunes position with this same theory, so the astronomy world was eager to see if Vulcan really did exist. Le Verrier's Vulcan from the previous year was not large enough to account for Mercurys orbital variation, but the scientific community speculated that it was one of several planets, or possibly an asteroid belt. Unfortunately these "planets" would only be able to be observed during a total eclipse, or when they passed in front of the sun.
In the following months several astronomers "discovered" a total of 24 "objects" orbiting the sun in two distinct orbits. One belt with an orbital period of 38 days, and another with a period of 26 days. Most of these objects were observed by Professor Wolf at the Sunspot Data Center in Zurich, but several other astronomers backed up his claims.
In the years following many astronomers played "hide and go seek" with Vulcan. Nothing was observed during the total eclipse of 1860. Then a "round spot" was found in 1875 that seemed to be a part of Wolf's group of objects. This "dot" was photographed from two different locations in Europe. During the solar eclipse of 1878 observers saw not one but two "small planets", but neither of these "planets" had any relation to all the previous Vulcans that had been observed.
Then no one saw Vulcan ever again. It was searched for during every eclipse for the next 40 years, but it was never again observed. The General Theory of Relativity later explained Mercury's orbital variations without need for any inner planets. But it was never explained what all those astronomers were looking at before, comets, asteroids, and stars have all been used as explanations. Modern theory suggests that Vulcan may have been an asteroid or series of asteroids that were not in any sort of regular orbit (and were not even inside of Mercury's orbit at all).
Today the whole idea of Vulcan has been shelved. The last time anyone seriously considered the idea of an intra-mercurial planet was in 1971 when several objects were observed close to the sun during an eclipse. These were later thought to have been faint comets that were falling into the sun.