The universe is expanding. In the early 20th Century, an astronomer called Edwin Hubble made the discovery that distant stars emitted the same light as close-up stars (and the Sun), except they appeared more red. What's more, the more distant the star, the redder the star appeared. Hubble reasoned that, due to an effect called the Doppler shift, all the stars are moving away from us and that the entire universe is expanding. (Things which move away from you look slightly red, in the same way that a car zooming away from you makes a slightly lower-pitched sound.)
Since space is expanding, there must have been a time in the past (about fifteen billion years) when the universe was small. Very small. Infinitely small.
What does this mean for us? Will the universe just keep expanding? Or will it eventually come back into a Big Crunch? Well, there are two forces at work here. One of these is the expansion force caused by the Big Bang. It (obviously) makes the universe expand. The opposing force is gravity. It will pull the universe back to whence it came.
If the force of gravity is exactly equal to the force of expansion, then the universe will continue expanding forever. If there's more expansion force, then the universe will more than expand forever. (Best not to think about that too hard.) But if there's more gravity, then eventually the universe will collapse back into a very small region--the Big Crunch (or the gnaB giB). And then maybe we'll have another Big Bang. We don't know.
The geometry of the universe is related to this. If there's equal amounts of expansion and gravity, then the universe is flat in all directions, and goes out to infinity in all directions. If there's more gravity, then the universe is said to be closed and finite--if you travel in some direction, then you'll end up where you came from after a while. If there's more expansion, then the universe is open... it extends infinitely in more than all directions. (Best not to think of that either.)
Scientists like to think that the force of gravity is equal to the force of expansion. It creates a nice flat universe and makes their calculations easier. But in fact, the current evidence is that our rate of expansion is increasing, not slowing down at all, and we're destined to expand forever...
Well, how can we know if we have enough gravity? Gravity is proportional to the weight of an object, and we know how much things weigh. So we look for all the stuff in the universe, and from that calculate its weight, and therefore its gravity. Unfortunately, it's hard to see objects in the universe... we can only see bright things like stars, and they only make up about 10 percent of the critical mass we need. The things we can't see, we call 'dark matter', for obvious reasons.
Some other evidence for the Big Bang is the cosmic microwave background. Two scientists, by names of Penzias and Wilson, picked up some microwave signals in their antenna receiver. They couldn't work out where it came from, but they reasoned that the microwaves (which have a long wavelength) could have been visible light from the Big Bang, but from a long time ago and so very heavily redshifted. This is consistent with the Big Bang theory.
Another astronomer called Fred Hoyle doesn't like this theory. He coined the term 'Big Bang' because he thought it was silly. Instead, he proposed the Steady State Theory, which says, the universe has been expanding since forever, and new matter is constantly being created to help balance out the density. Nobody believes him because the evidence for the Big Bang is much stronger.
There are heaps of questions still unsolved, such as, where did all the Big Bang material come from? What happens before the beginning of time? What happens when you get a universe of zero size and infinite density? Is it the same as a black hole? We don't know. The branch of astronomy which deals with this is called cosmology, so become a cosmologist and find out!