Discovered in 1930 by Clyde Tombaugh, Pluto was classified as the ninth planet in our Solar System. This would be taught in classrooms for nearly a century, until science began having a better understanding of what shape our Solar System really has and how it works.
In 2006, Pluto was declassified as a Planet because of new definitions set forth by the IAU (International Astronomical Union), which listed three mandatory rules in order for a solar object to be classified as a "planet" in our Solar System. In order to be classified as a planet, the object should:
1) Be in orbit around the Sun,
2) Has sufficient mass to assume hydrostatic equilibrium (a round shape), and
3) Have "cleared the neighbourhood" around its orbit.
Pluto failed to meet the standards of rule three, which essentially means that the solar object must have sufficiently much larger mass than its neighboring bodies. For example, Earth's mass is much larger than the moon and other neighboring bodies that are around a radius close enough to Earth.
The reason why Pluto failed to meet the third rule is essentially because of the Kuiper Belt. The Kuiper Belt is a large "ring" of asteroids and other foreign objects that orbit our Sun, but are outside of the general Solar System's vicinity. The green marks below identify objects that are in the Kuiper Belt:
Contrary to what astronomers originally thought, Pluto was not orbiting the Sun on its own. It was among thousands of other objects, some near the mass of Pluto, all orbiting the Sun in the Kuiper Belt. By studying the trajectory of these objects in the Kuiper Belt, astronomers were able to understand that Pluto was simply a larger mass that stood out among the other Kuiper Belt objects.
A large reason why the search for a 9th planet was started was due to the fact that Neptune's orbit is not currently completely explained. This is because Neptune's orbit is more skewed than it should be due to Uranus--meaning that something with a large mass--further out of Neptune's orbit is acting on Neptune, changing its trajectory. This search for this "Planet X" continues to this day. Scientists have identified a few new "plutoids" such as Eris and Sedna. However, they do not have the mass necessary to cause the skew in Neptune's orbit.
Pluto was simply a misidentification by scientists in the search for Planet X which is responsible for the skew in Neptune's orbit.