The Big Bang theory states that around 13.7 billion years ago the universe was condensed into an incredibly small, hot, dense "ball" of space and time called a singularity. At that time there was no physical matter in the universe.
The name "Big Bang" is somewhat of a misnomer, since the universe simply expanded and didn't literally explode (and it certainly didn't make a "bang" sound). As the expansion continued, the universe cooled, eventually reaching a point at which particles of matter could "freeze out" of pure energy (see Wikipedia:Mass–energy equivalence) and collide with each other to form the first simple atoms. These atoms continued to collide creating progressively "heavier" elements through the process of nuclear fusion. Over billions of years, these particles combined to form "clouds" of matter which further condensed, because of gravitational attraction, into stars and planets. (See Wikipedia:Physical cosmology for much more detail.)
The history of the universe can be described in some detail back to the instant approximately 10-43 seconds after the big bang. What occurred in the first 10-43 seconds (the Planck epoch) is not known and difficult to theorize, due to interactions between the theories of gravitation and quantum mechanics.
Creationists often object to the Big Bang theory on the grounds that it removes God's hand from creation. A common Creationist argument against it is the question "What caused the Big Bang?" and the closely related question "What happened before the Big Bang?"
There is a common misconception that the Big Bang means that the universe "came from nothing." Creationists use this as a launching point to claim that without introducing God, the first law of thermodynamics would be violated.
The Big Bang theory does not say that the dot came "from nothing"; it simply postulates the existence of the singularity and then proceeds from there. The answer to the question of what came before the Big Bang is simply "No one knows yet."
This lack of knowledge cannot be used to assert the existence of a God, however, as one can always ask "Who created God?" Since Creationists typically believe that every effect needs a cause they assume that the universe needed one too. See the Cosmological argument.
Speculations on the pre-Big Bang universe
- The oscillatory universe is the hypothesis, attributable to Richard Tolman from 1934, that the universe undergoes an infinite series of oscillations, each beginning with a big bang and ending with a big crunch. After the big bang, the universe expands for a while before the gravitational attraction of matter causes it to collapse back in and undergo a bounce. (This theory has declined in popularity since 1998, when astronomers reported evidence that the acceleration of the universe's expansion continues unabated.) 
- Cosmological natural selection is a speculative hypothesis proposed by Lee Smolin. Smolin speculates that every black hole might contain another universe inside it. Thus, our universe might be a black hole inside another universe. Each universe shares properties and fundamental constants with its "parent" universe, but may be slightly different. Thus--according to this theory--universes evolve over time, and the ones that are particularly well suited to produce black holes are the ones that thrive.
- The multiverse hypothesis suggests that there are already multiple parallel universes, generated in a meta-universe.
- Strictly speaking, time is not linear. When two people are moving relative to one another at speeds which are a significant fraction of the speed of light they will measure markedly different amounts of time passing compared to one another. Relativistic theories imply that time itself erupted from the big bang; the idea that an event could occur before time itself would under this pretense be logically invalid. (This theory suggests that the universe is infinitely old, and thus had no "origin")
None of these concepts have been conclusively demonstrated, but they do illustrate that God isn't the only possible answer (See Wikipedia:Cosmogony for more information).