Nuclear fission is a process in which the nucleus of an atoms splits into smaller parts (hence, fission). This usually occurs as a nuclear reaction, or radioactive decay. The splitting results to the release of various particles. Examples of particles produced or emitted or released are neutrons and photons.
The process of nuclear fission was discovered in 1938 by two physicists and two chemists - Lise Meitner, Physicist; Otto Hahn, German Chemist; Fritz Strassmann, German Chemist; and Otto Robert Frisch Austrian-British Physicist.
This discovery followed a historical successive discoveries involving subatomic particles. Fermi tried bombarding uranium with neutrons, which Chadwick discovered two years prior. He claimed to have created two new elements of lower atomic number. His research wasn't highly accepted, but was replicated by Hahn, Meitner and Strassmann. They basically got the same results, and the correct interpretation that the nucleus of uranium was split in half was provided by Meitner and Frisch. Frisch is also credited for suggesting the name "nuclear fission" to the process, relating it to fission in as it is used in biology.
Hence, scientists Hahn, Frisch, Meitner, and Strassmann, are all credited for the discovery of nuclear fission.
Though, again, the entire process of discovery was through piling up of accumulated data from various laboratories.