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Regular bullets (non-bonded) are made by pressing a lead core into the jacket under high pressure, and closing the jacket around the top. It is done under such pressure (called swaging) that the lead flows into all spaces. While the lead perfectly fits the inside of the jacket, it is not actually attached to it. The jacket is just that -- a jacket wrapped around the outside of the bullet like your coat around you.

With a bonded bullet, the lead is (at least in one process) melted inside the jacket and a chemical agent added that causes the lead to actually attach itself to the jacket. When it cools, the final forming is then done as with a normal bullet. So to finish the analogy, it's like sewing one side of velcro to the inside of you jacked and the other side to your shirt so when you put the jacket on it's physically attached to you.

The result is that when the bullet opens, the heavy lead core does not split apart from the jacket. If part of the jacket is ripped off, it must take a chunk of the lead with it since they are attached. The result is the bullet holds together a lot better.

The downside is that the bullet is tougher, and on thin-skinned game or shots that don't hit anything hard the bullet won't expand as well.