It's called dynamic text. Do a Google search for Dynamic Text After Effects and you'll come up with a bunch of tutorials and examples.
This is much easier and faster to do in AE than it Flash. You'll need a good understanding of 3D Cameras, parenting, and linear and bezier keyframes. Pre Planning is key to making this kind of project not only work but completing it in a reasonable amount of time.
Thanks for your response Rick, you said it is easier to do in AE. Is it because you have to use 'code' in Flash to make the 3D things happen? I still can't decide what program to use, I have only little knowledge in AE, Flash and Apple Motion and I prefer Flash in general.
It's easier in AE because you can arrange all your type on a big canvas. I usually do the text layout in Illustrator. Then you attach the camera to a null and drag it around the stage. In Flash you are moving things on and off the stage. I think it's just more labor intensive.
Here's a tutorial that covers one technique. It's similar to the workflow that I use most of the time except that I lay out my type in Illustrator.
I originally called this style Dynamic Text. It's actually more often called Kinetic Text Animation. A Google search will bring up a bunch of tutorials. While you can do this in flash I find that it's easier in AE, but if you're very comfortable in Flash to ahead and work there. I've never liked trying to sync audio to animation in Flash.