The center of pressure is the point where the aerodynamic moments go to zero. If you sum about this point, the pressures produce a pure force and zero moment.
If the center of pressure is not located at the CG, the aircraft will rotate. Therefore, if an aircraft is trimmed, then the CP has been positioned to coincide with the CG. We never think of it this way, but it has to be true.
*This argument ignores the moment caused by the engines. And technically, the CP applies only to pressure, so it also ignores the moments caused by viscous shear forces. But, the inviscid pressure forces dominate an aircraft -- they are what we use to hold it up (lift) and to control it (roll, pitch, yaw moments).
The CP changes with angle of attack -- and also with control surface deflection. So, trimming an aircraft requires simultaneously finding the angle of attack and control surface deflections to achieve the desired lift coefficient (target trimmed airspeed) and zero moment. Fortunately, for the most part, these equations are linear and we can solve this with a simple system of two linear equations. Typically written as a 2x2 matrix equation.
Since the CP moves with angle of attack, it is not useful for the stability discussion. Instead, that focuses on the aerodynamic center or the neutral point.
AC and NP are pretty much analogous terms. We usually use AC when talking about 2D airfoils and also 3D wings in isolation. We usually use NP when talking about a 3D aircraft as a whole. The NP is a weighted (by area) average of the AC's of all lifting surfaces (wing, tail, canard) for an aircraft.
The AC and NP are defined as the point where the moment is constant as you change angle of attack. This is only true in the linear, well-behaved region of flow. When you approach stall (for example), this falls apart.