Atmospheric nuclear explosions are associated with mushroom clouds, although mushroom clouds can occur as a result of large chemical explosions. It is also possible to have an air-burst nuclear explosion without these clouds. Nuclear explosions produce radiation and radioactive debris.
The first nuclear weapon was detonated on July 16, 1945, when the United States tested a Plutonium type implosion-device called Gadget in Trinity Site, New Mexico. The second use of a nuclear weapon was in combat on August 6th when the United States dropped a uranium type device called Little Boy on the Japanese city of Hiroshima. The United States also bombed Nagasaki three days later with a similar bomb to Gadget, called Fat Man, making it the last use of nuclear weapons in combat. These bombings resulted in the immediate deaths of around 120,000 to 220,000 people while more died over time due the nuclear radiation.
Power plant accidents such as those at Chernobyl and Fukushima were caused by steam and hydrogen explosions, not nuclear explosions. The fuel in a nuclear power plant is not enriched enough to create a nuclear explosion.