According to the Nobel Foundation, Henri Becquerel discovered radiation in 1896 while experimenting with uranium salts. He believed the uranium absorbed energy from sunlight, since the salts fluoresced in the sun, and then radiated that energy out over time. An accident disproved his hypothesis, and proved to Becquerel that the energy the uranium salts released needed no external stimulus.
Becquerel's experiment involved using uranium to fog photographic plates. He would wrap the plates in order to protect them from sunlight, then place them with a sample of uranium in the sun. He believed the sun excited the uranium, and the uranium sample then radiated that energy back out into the photographic plate. However, during a cloudy stretch, he stored his equipment for several days, only to discover that the uranium had emitted plenty of energy without outside stimulus.
At first, Becquerel thought he had discovered another emitter of X-rays, which were identified the previous year. However, X-rays could not be affected by magnetic fields, whereas the energy released by the various radioactive substances he experimented with were subject to magnetic attraction. Becquerel did not pursue this experiment further, and the husband and wife team of Pierre and Marie Curie built on his experiments. Becquerel and the Curies received the joint Nobel Prize in 1903 for their discovery of radiation.