Before Pluto, we knew of eight other spherical objects that revolved around the sun. Five of them were known to ancient astronomers: Mercury, Venus, Mars, Jupiter, and Saturn. Later we learned that the Earth was a spherical body that also revolved around the sun, so it was a planet too. Uranus was discovered in 1781 and Neptune in 1846, bringing the total to eight.
Clyde Tombough discovered Pluto in 1930. He lumped it in with the planets, and that gave us the nine planets that we all learned. Until 2006. The problem was that by 2006 two facts had become clear:
Pluto had characteristics that were different from the other eight planets.
There were a very large number of Pluto-like objects in the outer solar system.
If we excluded objects like Pluto from the list of planets, then there would be eight planets. If we included objects like Pluto, then we would have to include Ceres, Eris, Sedna, Haumea, Makemake, Orcus, and many others. So the number of planets wasn’t going to stay at 9. With the discovery of so many new objects in our solar system, the International Astronomical Union (IAU) the international body that is responsible for the names and classifications of astronomical bodies decided that it was time to sit down and figure out exactly what the classification of "planet" means. In 2006 they announced that a planet was something that had these characteristics:
Orbits the sun.
Spherical. (Actually, the rule said that the planet had to be in hydrostatic equilibrium, but for practical purposes, that meant that the planet’s gravity had pulled it into a sphere.)
Has cleared its orbit.
The last one needs some explanation. When the solar system was in its infancy, the solar system was full of debris. There were countless bits of rubble in the Earth’s orbit. However, over time the Earth either absorbed all of this material or flung it into a different orbit. Today, just about the only thing that occupies the Earth’s orbit is the Earth. The Earth has cleared its orbit. So have all the other planets.
Pluto doesn’t meet all three of these requirements. It orbits the sun and is spherical, so it meets those first two requirements, but it hasn’t cleared its orbit.
Pluto, Ceres, Sedna, Eris, and others are now considered to be dwarf planets. A dwarf planet is a body that meets the first two requirements, but not the third (i.e. hasn’t cleared its orbit).
Simply, the IAU decided that it was useful to classify Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune as planets, because they have certain characteristics in common. Meanwhile, Pluto, Ceres, Sedna, and the other similar objects are now classified as dwarf planets, because they have certain characteristics in common.