What is the difference between centre of pressure, aerodynamic centre and neutral point?
Any body moving in a fluid experiences pressure forces over its surface. The concepts of center of pressure, aerodynamics center and neutral point are useful in understanding the effects of these forces. Let's take an airfoil moving in air with subsonic flow attached to the body.

Center of Pressure The center of pressure is the point where the total sum of a pressure field acts on a body. In aerospace, this is the point on the airfoil (or wing) where the resultant vector (of lift and drag) acts.



Source: avstop.com

As the airfoil angle of attack changes, the pressure field changes. Due to this, the center of pressure changes with variation in the angle of attack. In the airplane's normal range of flight attitudes, if the angle of attack is increased, the center of pressure moves forward; and if decreased, it moves rearward.



Source: www.flightlearnings.com

As the center of gravity (the point where the weight of the body acts) is fixed, this movement of center of pressure affects the  stability of the aircraft.

Aerodynamic Center The resultant (or the pressure forces) also cause a moment on the airfoil. As the angle of attack increases, the pitching moment at a point (for example, the center of gravity) also changes. However, the pitching moment remains constant at a particular point, which is called the aerodynamic center.



Source: www.theairlinepilots.com

For symmetric airfoils in subsonic flight the aerodynamic center is located approximately 25% of the chord from the leading edge of the airfoil. This point is described as the quarter-chord point. 

Thus the aerodynamic center does not change with variation in angle of attack. Due to this, the aerodynamic center, rather than the center of pressure is used in the analysis of longitudinal stability.

Neutral Point Consider the whole aircraft. The lift of the wing acting through the center of pressure is in front of the center of gravity of the aircraft. This causes a destabilizing motion (increase in lift to increase in angle of attack causes a nose up moment, further increasing angle of attack). This is counteracted by the moment produced by the lift of the horizontal stabilizer (which is a small wing), acting behind the center of gravity.



"AirStability" by User_A1 - SVG derivative of Public Domain work by G J Vigurs. Licensed under GFDL via Commons.

As the center of gravity is moved forward, the stability of the aircraft increases (as the main wing lift arm is reduced) and the aircraft is statically stable. 



Source: quest.arc.nasa.gov

As the center of gravity is moved aft, the main wing lift moment arm increases and the aircraft stability decreases. The aircraft is said to be statically unstable.



Source: quest.arc.nasa.gov

As the aircraft is stable when center of gravity is in the nose and unstable when center of gravity is in the tail, there is a position in the middle where the aircraft is neither stable nor unstable i.e the stability is neutral. This point is called the neutral point. The neutral point is fixed for a particular aerodynamic configuration of the aircraft.



Source: quest.arc.nasa.gov

Thus, 


Center of pressure of an aircraft is the point where the Lift acts. 
Aerodynamic center is the point in the wing where the pitching moments are constant.
The neutral point is where the center of gravity of the aircraft is neutrally stable