When I start Skyward Sword (its the only MotionPlus game I have, others that use it are probably similar) it always asks me to lay the MotionPlus controller on a flat surface for a bit before it starts, then it asks me to point the controller at the center of the screen and press A. I don't have to do this for games that don't use MotionPlus, so what exactly is happening here?
The original Wii controller contains just an accelerometer, so it is only able to capture simple movements, like up/down. Wii MotionPlus contains a gyroscope that captures more complicated movement. From Wikipedia:
The [Wii MotionPlus] incorporates a dual-axis tuning fork gyroscope, and a single-axis gyroscope which can determine rotational motion. The information captured by the angular rate sensor can then be used to distinguish true linear motion from the accelerometer readings. This allows for the capture of more complex movements than possible with the Wii Remote alone.
Laying the controller flat initializes the gyroscope so that it knows where position 0 is, so to speak, which then allows it to detect when and how far it moves away from that position.
The MotionPlus needs to calibrate itself so it can accurately how and when you're moving it.
This is necessary due to the specific technology used for the MotionPlus, which is why you don't get it on non-MotionPlus games - the plain Wii Remote uses an accelerometer to detect movement, whereas the MotionPlus uses gyroscopes.