I found when exercising (through Kinect, pilates video, etc) that I am asked to 'engage my core'. How do I do this? How do I know when I am truly engaging my core rather than just sucking in my stomache?
The core, as it is referred to, consists of the muscles of your torso, such as your pecs, lats, psoas, abs, traps, obliques, etc. When you are asked to "engage" these muscles, this means utilizing them for stability and smooth motion. Rather than just flinging yourself wildly and letting the forces of nature work on you (centrifugal force, gravity, etc), you need to "engage your core" to produce smooth, stable movements.
Think of raising your arms above your head and lowering them back to your sides. You have to use your traps to raise the arms, but you can just let gravity take over to lower them and let them flop back to your sides. To get the full benefit though, you would want to raise your arms slowly, engaging muscles to fight against the raising of your arms, although not to make it impossible to raise them. Lowering them, you would want to utilize muscles to lower them, and also fight against the lowering at the same time.
This engagement works two or more sets of muscles simultaneously, and if you follow the directions you have, taking this sort of "fight" into account, you'll get a much more well-rounded and beneficial workout.
Here's one simple visualization technique that engages and stabilizes your core:
Imagine a straight line running through your body connecting anus to belly-button. Now, without sucking in, try to make that line as short as possible.