Short answers, pulled from the story.
French physicist Jacques Babinet suggested incandescent clouds circling the star in 1846 and named one potential planet Vulcan after the Roman god of fire. Urbain Le Verrier later calculated an orbit for this hypothetical world based on discrepancies in Mercury's perihelion motion.
Edmond Modeste Lescarbault observed what he believed was a small black dot moving across the Sun on the 26th of March 1859 using a modest 3.75-inch refractor telescope. He sent a letter describing the event to Urbain Le Verrier who announced the discovery at the Académie des Sciences in Paris on the 2nd of January 1860.
Astronomers searched for Vulcan during total solar eclipses because they believed unseen gravitational forces were pulling Mercury off course due to its advancing perihelion. Observers like James Craig Watson and Lewis Swift claimed to see red planets or definite disks near the Sun but their coordinates failed to match known stars.
Albert Einstein published his general theory of relativity in 1915 which showed that spacetime curves due to the Sun's mass causing Mercury's perihelion to advance by exactly 43 arcseconds per century. This explanation matched observations perfectly while eliminating the need for any unseen planet inside Mercury's orbit.
The International Astronomical Union reserves the term 'Vulcanoid' for asteroids possibly existing within Mercury's path today. Despite searches using earth-based telescopes and NASA's Parker Solar Probe no such objects have been found.