Unit turn video:
http://www.youtube.com/watch?v=a-PtbrF6v4E
As larry10's is pointing out above, the term "unit turn" is usually meant to the turn away with the shoulders, thus rotating the whole upper body, as the ball is approaching. The above video demonstrates the "unit turn" of rotating the upper body back, then the the reversal of the "unit turn" as you begin to swing forward.
Once the upper is body is "coiled", it then can be uncoiled with the rotational force powering the shot, not just the arm.
[note: I started writing this reply before
In the video of Federer you provided, his feet art not parallel to the baseline.
His feet are more at a 45 degree angle to the baseline, like in the middle hitting sequence below:
In this middle sequence of photos, he has already completed his unit turn (like he has already completed it in the video) and so this sequence just shows him uncoiling the upper body as he hits the shot.
But Fed, like all pros, and even the best players in your area, modify his foot positioning depending on the situation.
So in the first photo sequence above his feet are parallel to the baseline. This is often the feet positioning for a return of serve, where all it takes is a quick turn of the shoulders (i.e. unit turn) and he can then unwind with power, without having had to move his feet.
And in the bottom most sequence, he appears to have had to move somewhat to his right and move in for a low ball, so he has to position his feet perpendicular to the sideline. So there is less of a shoulder/unit turn backwards (since he is facing the sideline already), but he still rotates his shoulders/upper body back forward forcefully to power the shot.
When you play, don't worry about what your feet are doing after the hit. Get into as good as position as possible, do your unit turn which involves letting your weight sink down/bending your knees to do it optimally.
The reversal of the unit turn is synchronized with pushing hard off your legs to help you unwind, and add the power of your legs into the shot.
Your powerful leg pushoff may even take you up in the air, as Fed often does, and your own inate ability to land will just take over.
[Note: I started writing this reply before Fuzz Nation posted. It sounds like we are in complete agreement.]