Differenciate- the motion of "Nose-up trim" and "Nose-down trim"
There are two pitch inputs* in the cockpit: the yoke and the trim wheel. The effects add up - you can achieve the same pitch up moment by the yoke only, the trim wheel only, or a combination of the two. The pitch up or pitch down motion of the airplane is always the sum of the two.

*That is not a very correct statement, but for now let's put things that way for easier understanding.

The difference between the yoke and the trim wheel is that once you let go of the yoke, it returns to neutral. However the trim wheel stays where it is.

So let's say you want 2 units of "pitch up force" (and let's ignore what the actual unit in physics is, it is just a number) in a certain flying situation. You pull on the yoke until you reach the desired pitch. Great. However if you let go of the yoke, you will loose that pitch up force and the aircraft will start to pitch down. That is not what we want - we want the airplane to hold this pitch attitude, stable, for the next 10 minutes. If we move the trim wheel from the neutral position to 0.5 units up, the desired amount of pulling on the yoke is reduced - we still need to pull back on the yoke, but not as much - we only need 1.5 units of pitch up on the yoke because the other 0.5 units is provided by the trim wheel. This is what "relieve back pressure" is referring to. If you move the trim wheel to 2 units up, you can now completely let go of the yoke, and the aircraft is stable.


  Is the motion in opposite of when pushed down, nose up trim and vice
  versa to that motion?


I think you might be confusing the pitch input with the movement of the trim tab on the elevator, assuming that your textbook mentions it. On most general aircrafts, a trim tab is used for pitch trim control. A trim tab is essentially "an elevator of the elevator". The elevator needs to move up for the airplane to pitch up. But the trim tab needs to move down for the airplane to pitch up.

Do not confuse how the trim is achieved aerodynamically, with what you input in the cockpit. The trim tab is just one of many ways to control pitch trim. Airliners, for example, do not have a trim tab. However, the principal of operating the trim wheel is the same. It is just a large aircraft uses a different method to hold a stable pitch attitude.