Can time have a beginning?
At first this seems impossible. If time has a beginning, there must be a first event or at least a clustering of events near it. Surely something must have happened before them?
Einstein's cosmology of 1917 was the first of many ever stranger cosmologies to be devised on the basis of his general theory of relativity. Einstein's first universe was static in time. The cosmologies that followed, starting in the 1920s, were not. They portrayed space itself as continually expanding. We can think of Einstein's universe as a three dimensional analog of a two dimensional spherical surface, somewhat like a balloon. Then this expansion simply corresponds to the inflation of the balloon.
Here is a picture of this expansion. The universe is represented by a sphere and time advances up the screen. So the small universe of long ago grows up the page to the universe of the present.
Now imagine this expansion in reverse. As we look further and further back in time, the balloon gets smaller and smaller. In the typical cosmologies considered nowadays, not too long into the past the balloon would have shrivelled to nothing. At that point in our story, space would have ceased to be.
One might try to image times before that moment. But it would be futile, since there is no space associated with the time. Indeed there is something highly suspect about the moment at which the balloon shrivels to a point. Then the curvature of the space becomes infinite and the basic equations of Einstein's theory break down.
This first moment is not really a moment in time at all. It really amounts to a lower bound on our projections into the past. If we think of it as "time=0," then only moments with a time coordinate greater than 0 have physical meaning.
It is the beginning of time and is otherwise known at the "big bang."
To see why it is the "big bang", let us return our imagination to the forward direction and imagine what happens around the beginning of the expansion. Take any moment you like, as close as you like to the big bang. By choosing that moment closer and closer to the big bang, you can make space shrivel up as close to a point in size as you like. From that moment, everything--space and all its matter-- explodes outwards. All this happened not so long ago. It was around 10 billion years ago.