The Big Bang refers to the moment, about 13.8 billion years ago, when the observable universe began expanding from an extremely hot, dense state. It is not thought of as an explosion happening at a single point in existing space, rather, it is space itself that began expanding, carrying everything within it outward.
The evidence for it comes from several independent directions: the observed expansion of the universe today, the leftover Cosmic Microwave Background radiation, and the measured abundance of light elements like hydrogen and helium, all of which match what the Big Bang model predicts extremely well.
The Big Bang model describes how the universe evolved from that early hot state onward, it does not claim to explain what caused it or what, if anything, came before. That earlier question sits at the edge of current physics, and answering it would likely require a full theory of quantum gravity that does not yet exist.
The name itself was coined almost by accident. Fred Hoyle, an astronomer who favored a rival steady state theory, used the phrase ‘big bang’ dismissively on a 1949 BBC radio broadcast, and the name stuck even as the evidence increasingly favored the very model he was mocking. In the minutes after the Big Bang, the universe was hot enough to fuse hydrogen into helium and trace amounts of lithium, a process called Big Bang nucleosynthesis, and the predicted ratios of those elements match what telescopes actually observe with remarkable precision. It would take hundreds of millions of years more before the first stars switched on and began forging heavier elements, a period astronomers call the cosmic dark ages, since no light sources existed yet to illuminate the young universe.
The rival steady state theory, which held that the universe had no beginning and simply expanded while continuously creating new matter to keep its density constant, lost most of its support after the accidental discovery of the Cosmic Microwave Background in 1965. That faint radiation was exactly what the Big Bang model predicted should exist as leftover heat from a hot early universe, and the steady state theory had no comparable explanation for it.