The Big Bang is the popularly accepted cosmological model of the universe.
The idea is that the universe has expanded from a primordial hot and dense initial condition at some time in the past and continues to expand to this day. The framework for the model relies on Albert Einsteinâ€™s General Relativity as formulated by Alexander Friedmann. After Edwin Hubble discovered in 1929 that the distances to far away galaxies were generally proportional to their red shifts, this observation was taken to indicate that all very distant galaxies and clusters have an apparent velocity directly away from our vantage point. The farther away, the higher the apparent velocity. If the distance between galaxy clusters is increasing today, everything must have been closer together in the past. This idea has been considered in detail back in time to extreme densities and temperatures, and large particle accelerators have been built to experiment on and test such conditions, resulting in significant confirmation of the theory. But these accelerators can only probe so far into such high energy regimes. Without any evidence associated with the earliest instant of the expansion, the Big Bang theory cannot and does not provide any explanation for such an initial condition. The theory explains the universe since then.
A major success of the theory is its ability to account for the comparative abundance of the elements we find around us, which if you look beyond Earth is mostly hydrogen and helium. The observed abundances of the light elements throughout the cosmos closely match the calculated predictions for the formation of these elements from nuclear processes in the rapidly expanding and cooling first minutes of the universe, as logically and quantitatively detailed according to Big Bang nucleosynthesis and as described in Steven Weinbergâ€™s classic The First Three Minutes.
The term â€˜Big Bangâ€™ was apparently first coined by Fred Hoyle in a derisory statement seeking to belittle the credibility of the theory that he did not believe to be true. Ironically, Hoyle helped considerably in the effort to figure out the nuclear pathway for building certain heavier elements from lighter ones. Most scientists were fairly convinced by the evidence that some Big Bang scenario happened.