On Airbus aircraft, can the pilot control the aircraft when the autopilot is active? I am looking at the schematics of computerized control for Aibus aircraft and I do not understand the relation between the various control law and the autopilot. 

Under "normal law" can the pilot control the aircraft while the autopilot is engaged, or will control input from the pilot disconnect the autopilot automatically?

What is the meaning of "alternate law" for the autopilot? Will the autopilot be disconnected when shifting to alternate law?
Moving the flight controls above a certain threshold will disconnect the autopilot.
From the A320 FCOM (1.22.30 Auto Flight - Flight Guidance, emphasis mine):

AP DISENGAGEMENT
AP1 or 2 disengages when:

The pilot presses the takeover pushbutton on the sidestick.
The pilot presses the corresponding AP pushbutton on the FCU.
The pilot pushes on the sidestick harder than a certain threshold or moves on the rudder pedals above a threshold.
The pilot moves the pitch trim wheel beyond a certain threshold.
The other AP is engaged, except when localizer/glideslope modes are armed or engaged, or when the rollout or go-around mode is engaged.
Both thrust levers are set above the MCT detent and the aircraft is on the ground.
The aircraft reaches the MDA-50 feet (MDH-50 feet), or 400 feet AGL if not MDA/MDH, with APPR mode engaged and a non-ILS approach selected.
One of the engagement conditions is lost.

Furthermore, in normal law with all protections available, the AP will disconnect if:

High speed protection is active;
Angle-of-attack protection is active (α prot +1° is reached);
Pitch attitude exceeds 25° up, or 13° down, or bank angle exceeds 45°;
A rudder pedal deflection is more than 10° out of trim.


The different control laws act in the F/CTL COMPUTER in the following diagram:

(source: FCOM 1.27.10 Flight Controls - Description)
They are independent of the autopilot and the protections are available regardless of whether or not the autopilot is engaged.