Short answers, pulled from the story.
Vulcan was a proposed planet that some pre-20th century astronomers believed orbited between Mercury and the Sun. The name came from the Roman god of fire and was suggested in 1846 by the French physicist Jacques Babinet.
The French mathematician Urbain Le Verrier predicted Vulcan's existence after finding unexplained peculiarities in Mercury's orbit, publishing his findings in 1859. He had earlier used the same method, tracking orbital disturbances, to predict the existence of Neptune in 1846.
Lescarbault, a French physician and amateur astronomer, said he observed a transit of Vulcan on the 26th of March 1859. He reported the sighting to Le Verrier in a letter dated the 22nd of December 1859, and Le Verrier announced the discovery to the Académie des Sciences in Paris on the 2nd of January 1860.
Astronomers abandoned the search after Albert Einstein's 1915 theory of general relativity explained Mercury's orbital peculiarities without requiring an unseen planet. The theory showed the effect came from the curvature of spacetime caused by the Sun's mass, matching the observed 43 arcseconds per century precisely.
During the total solar eclipse of the 29th of July 1878, James Craig Watson observed from Separation Point, Wyoming, and Lewis Swift observed near Denver, Colorado. Both reported seeing a Vulcan-like object close to the Sun, though their coordinates did not match each other or any known star.
Astronomers William Wallace Campbell and Charles Dillon Perrine of the Lick Observatory concluded the observational search in 1908, after photographic expeditions to the solar eclipses of 1901, 1905, and 1908. The International Astronomical Union later reserved the name Vulcanoid for any small asteroids that might exist inside Mercury's orbit, though none have been found so far.