If you've got a bunch of footage and you want to cut back and forth between clips of that footage to tell a story, then use Premiere Pro.
If you want to create motion graphics, create visual effects, and composite one layer of graphics/footage over another, then use After Effects.
You can do many of the same things in each program, but you'll quickly learn that each has its strengths.
I suggest starting here to learn After Effects, including what it's good for:
This will more or less be my first foray into AE.
Premiere is a video editor. After Effects is a compositing system. You must be comfortable with the diference before you can even start to grasp how the applications work separately. Then you can begin to appreciate how they work together.Similarly, your suite includes and audio editor and a DVD authoring application. Both of those basic functions can be initiated in Premiere but, if you are creating DVDs, you must move your edited program to Encore. If you need ultrsophisticated audio processing and 5.1 mixing, you will move your audio tracks to, umm, what's it called? Soundbooth.
Don't jump into AE as a lark, it will frighten you. Open the manuals, slog through a few of the hundreds of free tutorials. Don't ask beginner questions until you're no longer a beginner. I've been using AE for more than 19 years and I still don't get i. After Effects is huge. And deep. And ridiculously complex. It's truly scary. Which is why we often say, learn how to edit video before you try to learn how to do effects for video.