What happens when two different control inputs are commanded in an aircraft with two control columns?
It depends on the aircraft. In any aircraft where the two control columns are linked, there's only one input to command the control surfaces. While most smaller aircraft are connected directly through the control surfaces, Boeing publicizes that on the 787 - a fly-by-wire aircraft - the control columns "[a]re cross-linked between pilots to reduce potential confusion."

On Airbus aircraft (and probably others), the controls are not linked: there is no indication on one pilot's controls of the other's position. The computer handles this by adding the two inputs: opposite commands will cancel each other out, while both pilots commanding the same input will increase the total commanded input. Some Airbus aircraft have a feature which makes an announcement of "DUAL INPUT" when there is input on both sidesticks simultaneously. As of January 30, 1997, this was "an installation" (optional feature, I believe) on the A320 and under development for the A340.

Airbus aircraft also have a Priority Takeover button, which lets one pilot take control and have the airplane ignore inputs from the other sidestick. As Farhan noted, good CRM (Crew Resource Management) is also very important to ensure that only one command is given at once.

Some aircraft have linked controls, which can be separated if necessary. On the 767, if sufficient force is applied, the elevators will "split": each control column will control one of the elevators. The pilot who is stronger or better prepared will have more control over the aircraft. There's also the EMB-145 and -135:


  The roll input is ... linked at the control column[.] Each yoke independently controls its own aileron -- the CA yoke only flies the left aileron and the FO yoke the right aileron. It is only through the control column linkage that each yoke can control the whole airplane. It is done this way so that if there is a problem with a cable jamming, the control interlock can be disconnected and at least half control regained.