I will yield this knowledge to Rick Stowell in his Emergency Maneuver Training book and slightly tweak it to how I like to explain it.
Forget what you think you know, forget about taildraggers and conventional gear, forget it all except: when a force is applied to a rotating disk it acts 90 degrees from the place it was applied. We can build off of that.
The propeller is, for all intensive purposes, a rotating disk. When we pitch up, down, or apply rudder left or right we put a force on that disk.
Let's make a picture out of it, go get a stick, bat, broom, golf club, or any other stick type of object. Yes, I am serious, go get one before you continue reading. Now with your 'stick' sit in the chair and hold it out in front of you. Put your right hand on the top and left hand on the bottom, holding it vertically.
What do we know about the propeller when we are sitting in the aircraft? We know it rotates to the right? If you have to, spin the stick to the right to imagine that and then reposition it vertically the way you started. Let us apply some forces and see what happens:
Pitch Up: With your right hand, pull back on the stick and with your left push forward. Now rotate the stick 90 degrees to the right (horizontal), your right hand will now be on the right side of your body, pulled back, with your left hand to your left and pushed forward. The result in a pitch up is a right yawing tendency, NOT a left yawing tendency.
Weird, why do they call it a left turning tendency when it is a yaw and to the right? Introduce the tail dragger! A tail dragger, as it begins its takeoff roll will raise the tail (pitch down) which we will now discuss.
Pitch Down: Go back to the vertical starting point with your stick and this time push your right hand forward and pull your left hand in (your right hand is still on the top). Now rotate 90 degrees clockwise again and notice your right hand on the right side is pushed forward and your left hand on your left side is pulled back. The result, a yaw to the left.
What does this mean to the aerobatic pilot? Well each time they push forward they must apply some right rudder and each time they pitch back they must apply some left rudder. Learning to do this accurately is what many don't realize, people think an aerobatic guy just does a lot of yanking and banking with a really cool airplane. Not the case, and it even goes further than this.
Let's look at the application of right or left rudder. For this you need to turn your stick horizontal and grab it with both hands again, right on the right and left on the left. It is easiest if your knuckles face the ceiling, palms facing the floor.
Right Rudder: Pull back on your right hand and push forward on your left to demonstrate applying right rudder. Now move this force 90 degrees clockwise and notice your right hand, on the bottom, is pulled back. Your left hand, on the top, is pushed forward. The result, a pitch down moment when right rudder is applied.
Left Rudder: Return to horizontal and pull back with your left hand, push forward with your right. Rotate 90 degrees clockwise again and notice your left hand, on top, is pulled back. Your right hand, on the bottom, is pushed forward. Result, a pitch up moment when right rudder is applied.
An aerobatic pilot will have to think about all of these possibilities. It is what makes the 8 point roll so incredible when you realize the precision that is involved. Their big engines exhibit a very noticeable yaw and pitch with application of pitch or yaw described above.
For the GA pilot, in a trainer aircraft this information can basically be ignored. A full forward or full aft pitch in your typical 172 will cause the ball to move about half to a full ball movement from one side or another. It is rare you will ever pitch that extreme. Your typical pitch will move less than a half a ball.
As you move into higher performance aircraft, if you make quicker pitch actions then it might be worth while to consider learning how to use the rudder with pitch. Just like adverse yaw while rolling though, it depends on how intense the pitch is. A smooth application of pitch in almost any non aerobatic aircraft will not need rudder.
Moral of the story? Apply pitch smoothly and unless you intend on being an aerobatic guy you can ignore gyroscopic precession. IMO it is an overly stressed term to private pilots that has little, if any, effect in the world they operate.
xcaliber: Torque is a left rolling, not yawing tendency. P-factor is a left yaw, but not nearly as significant as slip stream. Here is the list I use if it helps:
Torque = left roll (right yaw...adverse yaw)
P-factor = small left yaw
Slip stream = largest left yaw
Gyro Prec = In taildraggers a significant left yaw when the tail is raised. For trainers and most other GA aircraft almost not noticeable and depends on the force applied discussed above.