Perhaps there is some disagreement about terminology here. "Center of Pressure" and "Aerodynamic Center" refer to completely different concepts, both of which have utility in different contexts. For 2 dimensional wing sections I think the difference is pretty easy to understand.

Center of Pressure: The position along the chord of a wing section about which the pitching moment is zero. Linear analysis of thin wing sections predicts that the Center of Pressure will (in general) move as the angle of attack is changed. One exception to this is a symmetrical (uncambered) wing section.

Aerodynamic Center: The position along the chord of a wing section about which the pitching moment does not change as the angle of attack is changed. The pitching moment about the Aerodynamic center may or may not be zero. If it is zero, then the Aerodynamic Center and the Center of Pressure happen to be at the same point. Linear analysis of thin wing sections shows that the Aerodynamic Center is at c/4, or "quarter chord", and doesn't change with angle of attack.

Both concepts can be extended to 3D aircraft and thick airfoils. One thing that changes as you go to a 3D aircraft from a 2D wing section is that the Aerodynamic Center (usually) no longer falls at the quarter chord. Another difference is that the location of the Aerodynamic Center is no longer strictly independent of angle of attack. However, it remains fairly stationary even for fairly thick airfoils over a wide angle of attack range (as alasdair points out).

THE condition for static longitudinal stability is that the Center of Gravity is forward of the Aerodynamic Center. The location of the CG in relation to the Center of Pressure will determine the static tail load, but is unrelated to stability. So Capio777, I've got to agree with alasdair that Center of Pressure is not directly relevant to the stability discussion.