This question may sound naive but surprisingly it puzzles me.
A wing has a centre of pressure, does it mean that lift acts on point and not on the whole chord of an airfoil?
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Center of pressure is not a point "where the lift is most concentrated". Center of pressure has very little relation to how is lift distributed along the wing (you can have exactly same centre of pressure for infinite number of different distributions of lift).
The center of pressure is mathematical sum of all local pressure vectors along whole wing surface. But it is only, more-or-less, thought summing, this point has no special meaning in terms of stress distribution or so (at least no straightforward one).
You could as well continue to put all local pressure points into equation of movement and you would get exactly the same result. But equations with plenty small force vectors would be unnecessary complicated and if you do no need this level of complexity, it is easier to sum all forces in advance and use the summed value instead.
Note that if you would want, say, for example, calculate stresses inside the wing, you can not do like this. You would be forced to put each local pressure point into the analysis. But as long as you could assume whole wing to be solid (which is usually acceptable approximation for lot of simple problems), using single summed lift vector is easier.
If you simply sum all fractional pressure vectors, you get total vector of aerodynamic force. Which is enough to know how big force and in which direction acts on the airplane. But in order to be able to analyze rotational effects, you need to know in which point this total force acts (because same force vector shifted into different point of solid body will cause different rotation). And this is where center of pressure comes into the play, as it is the point where thought summed aerodynamic force should act in order to affect airplane (as a whole) by same translational and rotational effects as actual pressure distribution along the wing does.
Aerodynamic force (part of which is Lift) is nothing more than the sum of all the individual aerodynamic forces acting on the surface of the airframe. At each and every tiny little square micrometer of the aircraft surface, the air is pushing on the airframe. It is pushing with some force equal to the normal air pressure at that point, ($lbs /inch^2)$ times the surface area of that tiny section. All these tiny aerodynamic forces push on the aircraft surface in a direction normal (perpendicular) to the surface.
To make things simpler (for calculations and explanations), engineers, and flight instructors, artificially sum all of these tiny individual forces into one total force. This is done by adding (vectorially) all the individual forces up into a single total force vector, which, if it was the only force acting on the airframe, would, theoretically, create exactly the same total force and rotational moment that all the tiny little real forces actually produce. This total artificial force is then broken up, again artificially, into two components, one that is normal (perpendicular) to the flight path vector of the aircraft (This is what we call Lift), and one which is parallel to the flight path (This is what we call Drag). They both can be considered to act through the same point that the total force vector was calculated to act through.
The point that this total artificial force is considered to act through is called the Center of Pressure.
What's it for? This is the point in the airframe, where,. if it was locked down at that point, the aerodynamic forces would be balanced around it and would produce no rotational moment. i.e., like putting a pin through an arrow, or the weathervane on your roof, if the pin through the arrow, or the axis of the weathervane went through the center of pressure, then the wind hitting the vane would not cause a rotational moment and the weathervane would not spin and align with the wind. It would just sit there, because the force (actually, the moment) on one side of the Center of pressure would be equal to the moment from forces on the other side and they would cancel each other out.
Why is this important? because in free flight, there is no pin, or axis attached to something immovable. An aircraft is free to move unconstrained. When an aerodynamic force is applied to a body which is free to move unconstrained it will rotate about it's center of Gravity. So if the Center of Pressure is not co-located with the center of gravity, (which it hardly ever is, exactly) then any aerodynamic force on the body will generate a rotational moment (a torque) about the CG.