How many planets are there in the Solar System?
Short answer: Eight.
Long answer: According to the International Astronomical Union, there are eight main planets in the Solar System. In addition, there are a number of smaller dwarf planets and many asteroids, meteoroids, comets and moons.
The decision in 2006 to change the classification of Pluto from planet to dwarf planet was controversial (see Why isn't Pluto a planet any more?). However this wasn't the first time that astronomers argued over the exact planet count. Even when Pluto was first discovered, many thought that it shouldn't be called a planet.
Before 1800 there were six known planets: Mercury, Venus, Earth, Mars, Jupiter and Saturn.
German/British astronomer William Herschel discovered Uranus in 1781, making seven planets.
In 1801, Italian astronomer Giuseppe Piazzi discovered Ceres, the largest object in the asteroid belt. Ceres was listed as a planet for about half a century before being demoted to asteroid (and later, dwarf planet).
In 1846, French mathematician Urbain Le Verrier and German astronomer Johann Galle discovered Neptune. This brought the planet count to eight, the same as it is today.
Following his work on Neptune, Urbain Le Verrier suggested that a planet should exist between the Sun and the orbit of Mercury. Some astronomers added this hypothetical planet, called Vulcan, to the total count. However it was eventually found that Vulcan does not exist.
American astronomer Clyde Tombaugh discovered Pluto in 1930. Despite some opposition is was classified as a planet and the total count moved to nine. In 2006 Pluto was demoted to dwarf planet, or "trans-neptunian object", and we were back to eight planets.
In 2016, astronomers Konstantin Batygin (Russian/American) and Michael Brown (American) proposed the existence of a planet farther from the Sun than Neptune. This hypothetical planet has been dubbed (Planet Nine) and is still waiting discovery.
So, for the time being, the total number of planets in the Solar System is eight.