It is theorized that the universe is expanding, and does so much like the way a balloon expands when it is being inflated. This means that the distances between the galaxies in the universe are increasing with time. The use of the balloon model is only to help one visualize how a 3D universe is growing larger while generally maintaining its form. The following is an interesting link, about a very simple experiment with background theory that helps a student see how inflating a balloon is really very much like the way the universe is expanding:

Having never heard of the universal balloon before this question I wanted to make sure I was going to answer the right question.I believe it's referring to the idea where you take a balloon, put marks on it, and then blow it up and observe the change in distance between the marks you've made.

This illustrates one way astronomers believe the universe is changing, which is expanding uniformly everywhere, radially. So if you start with a small ball and the ball grows and grows, while staying in the same shape, that's how the universe might be expanding.

The balloon helps you visualize how two spots that start close together, can end up very far apart.