This cast is designed to get your nymph (or wet or streamer) on the bottom quickly & to keep it there during the drift. If you're wanting to fish the bottom, this is the cast to use because your fly will enter the water first - before your leader & line.
To perform the tuck cast you must move your knuckles & thumb of your casting hand in a horizontal plane. You don't want this cast going to one side or the other because you'll lose the momentum in the line.
Keep your thumb on top of your rod grip the whole time you're doing this cast. It should come straight over head & you want to keep your wrist straight with your thumb.
So, you basically keep your rod, hand, & wrist in a horizontal plane & make a normal back cast. Then, on the forward cast stop the rod abruptly at about the 10:30 position & squeeze the rod grip with your thumb pushing while pinky & ring fingers pull the grip towards you.
It will look strange at first as the line seems to fall onto the water & your fly comes almost straight down onto the stream. Remember, the point of this cast is to get your fly on the stream bottom quickly, & so it enters the water first - before your leader.
During the drift keep your rod tip up. This keeps too much slack line out of the mix & keeps you connected by feel to your fly. There is already some slack in the leader when the fly hits the water just by the nature of the cast, so don't allow too much or you'll be sure to miss quick strikes from the fish.
Some descriptions of this cast tell you to overpower the forward stroke. That is wrong. The key to this cast is to stop the rod & squeeze with the hand. That's it. Otherwise it's a natural, relaxed cast.
The higher you stop the rod (mixed with the amount of weight you use), the steeper the angle of which your fly enters the water. For deeper water stop the rod at a higher angle. For more shallow water, stop it a little lower. Just remember to keep your rod tip up as you drift your fly.
One hint: if you see that your fly is splashing down into the water too hard - making too violent of a crash landing - then you're probably overpowering the forward stroke. Your fly should look like it just fell from the sky as it enters the water, as if someone just dropped it from a a few feet above the water.
Here is a video of an angler employing a proper tuck cast. Watch as his leader seems to just fall onto the water. It might even look like a bad cast, but it isn't. It's exactly what he was after.