The Big Bang theory is science's best explanation of how the universe was created. The theory asserts that our entire universe was created when a tiny (billions of times smaller than a proton), super-dense, super-hot mass exploded and began expanding very rapidly, eventually cooling and forming into the stars and galaxies with which we are familiar. This event is said to have happened approximately 15 billion years ago. Rather than expanding outward into some preexisting vacuum, the event of the Big Bang was space itself expanding - perhaps at speeds greater than light. (While Einstein's theory of relativity forbids anything within space from travelling faster than light, it sets no limitations on how fast the fabric of space itself may expand.)
The Big Bang theory was originally developed in the late 1920s by Georges-Henri Lemaître, a Belgian Catholic priest and astronomer, an early advocate of solutions to the general relativity field equations which predicted our universe was expanding. (For cosmological theories to be taken seriously, they must pose possible solutions to Einstein's general relativity field equations.) Though the expanding-universe solution to the field equations was derived by the Russian cosmologist Alexander Friedman in 1922, Lemaître was the first to realize that a continuously expanding universe implies that at some point in the past the universe must have been much denser and smaller, even atom-sized.
The Big Bang theory is supported primarily by two major lines of evidence - first, the fact that all galaxies are rapidly moving away from each other (confirmed by Edwin Hubble in 1929), and secondly, the presence of the cosmic microwave background radiation, or the "echo" of the Big Bang. The cosmic microwave background radiation was not discovered until 1965, and up to this point, scientists were divided between the Big Bang theory and its rival, Fred Hoyle's steady state model, which asserted that the universe was expanding, but staying basically the same because new matter was continuously being created.
Since the late 1960s, the Big Bang theory has been the dominant explanation for the birth of our universe. Fred Hoyle's steady state model has been discarded. Most of cosmology since that time has consisted of modifications and extensions of Big Bang theory. Because physicists have not yet formulated a consistent theory that explains how gravity operates on extremely small scales (like those present at the instant of the Big Bang), cosmologists are unable to formulate theories as to what happened before about 10^-43 seconds after the Big Bang. Our universe may have originated as a pointlike entity with nearly-infinite density, or perhaps something else. Our mathematics, instruments, and scientific methodologies may need to be substantially improved before any further progress is made.