A charming history begun at the beginning of 20th century. If we want to understand how Clyde William Tombaugh got the discover, we have to start about 90 years before, approximately in 1840

Two astronomers, Urbain LeVerrier (France) and John Couch Adams (Great Britain), each on their own, based on Uranus observations, forecasted and tried to find a new planet "disturbing" the Uranus orbit. Simply, they saw Uranus in a certain position in the sky but Newton's laws disagreed with it. The only logical consequence was that Uranus motion was perturbed by something else, as bigger as it. After few years Neptune was discovered very close to the position calculated by the two astronomers (LeVerrier made more accurate calculations).

Well, observing Neptune and noticing that its motion was disturbed too (based on mass and size), the equation had to be the same

There must be a celestial body, as big as Neptune, that disturbs its motion

In according to this theory, Percival Lowell (USA) started several calculations and observations until his death in 1916, without finding the ninth planet. In 1929 the chairman of the observatory, offered a job to Clyde William Tombaugh, enthusiast about astronomy who has sent his planet drawing to him. The job was very difficult: find the 9th planet!

Tombaugh had a smart idea

Keplero tells me that the more you move away from the Sun, your speed will be slower. Then I cannot realize the motion with my eyes, but i need an instrument that helps me to find the moving point between the stars.

Next 11 millions of "points", 5 months and 75 squared meters of photographs, he got it on the 18th February 1930!

Suddenly an object of fifteenth magnitude jumped into my eyes. "Here it is," I said to myself.
An incredible emotion overwhelmed me: this would have been a historical discovery. I went straight to the manager's office.
Trying to control myself, I entered the office flaunting indifference. - Dr. Slipher, I found your Planet X

He realized that this planet had too small size to disturb the motions of Uranus and Neptune, so he continued observe to find another planet, but he never found it. When he came back to the observatory in 1945, after have served as a professor for the NAVY durin 2nd world war, he got fired, officially because of lack of funds. Probably, the real reason for his dismissal, was told by Tombaugh after few years:

He was jealous that i had found it for him

Pluto was considered a planet from the 1930 to 2006 when he got downgraded to another category of body, the "dwarf planet" and it was achieved and studied only in 2014 by the New Horizons probe. Part of Tombaugh's ashes are contained on the probe.

Here's an interesting video about the New Horizon's mission. Enjoy