What phenomenon causes aerodynamic tail-down force? Why will there be more downforce on the tail when an aircraft picks up speed? What happens if the aircraft goes too slow - does this means less downforce on the tail? Do the center of lift and center of gravity play any part in determining the downforce on the tail?
The phenomenon is called trimming.
This is the process by which the pilot adjusts the lift at the tail surface to shift the center of pressure right above or below the center of gravity and removes any stick force so the airplane will keep this attitude unaided.
If the center of gravity (CG) is right above or below the aerodynamic center (AC), the aircraft's longitudinal static stability is indifferent. Move it forward and stability increases, move it further back and the plane becomes unstable. The aerodynamic center is where the angle-of-atttack-dependent lift forces act, and in an indifferent configuration the lift coefficient on wing and tail will be equal. Increasing stability thus means to reduce lift at the rear surface, and if a lot of stability is desired, lift at the rear surface will become negative.
When the airplane is hit by a gust (change in vertical wind speed), the angle of attack at both wing and tail will change by the same amount. Now it is important how much lift changes relative to the lift before the gust. If the lift coefficient on the wing is larger than on the tail, the same amount of angle of attack change will cause a relatively smaller change of lift on the wing than on the tail, and the center of pressure will move such that it causes a correcting pitch moment.
By selecting the proper center of gravity location, the pilot can adjust the level of stability he/she desires. To stay at the desired attitude in flight, he/she now has to move the incidence of the empennage (or the elevator deflection) to a point where any pitch moment disappears. The consequence is a downforce on the tail if that is what it needs to trim the aircraft with the given center of gravity location.
On a conventional wing and tail configuration the center of lift is normally behind the center of gravity. This causes a pitch-down force, which is counteracted with the stabilizer's downward lift. As speed increases, the wing produces more lift, thus more tail-down -force is required to maintain attitude. On a rear-wing configuration the same is accomplished with a canard. The only difference is that the canard provides upward lift and so it adds to the total lift, contrary to a conventional tail just increasing drag.
The tail is a lifting surface just like the wings. Downforce is just the phenomenon of lift directed in the opposite direction. You can also see examples of downforce in things like the spoilers on a Formula 1 car. Going faster produces more lift and so more downforce.
What's unique about the tail is that it can be affected by the downwash of the wings themselves which comes into play when looking at design choices like a T-tail.
Zooming out more broadly, the reason you want downforce is longitudinal stability. On some planes and in certain configurations the tail produces positive lift instead of downforce, all in the name of balancing the plane, though those designs tend to be harder to fly.
You may notice that the tail is usually much smaller than the wings. This allows for better longitudinal stability as the tail experiences larger relative changes in lift (downforce) compared to the wings which provides a negative feedback loop that brings the plane back to equilibrium.
The location of the center of gravity relative to the center of lift matters because it produces a certain pitching moment that then needs to be counteracted by the force produced by the tail.