The critical engine is the engine that produces the most yaw when it fails. More yaw requires more rudder input to counter it, which increases drag and reduces climb performance. The critical engine is the one you don't want to lose.
The left engine is more critical (to lose) than the right engine because of asymmetric blade effect. This is something you may have heard of in a single engine too - part of the reason you hold right rudder in during a climb. Essentially the down going blade travels faster than the up going blade (assuming an aircraft nose up attitude). This results in the down going blade producing more thrust than the up going blade. As the down going blade is producing more thrust, the thrust vector from the engine is offset from the hub in the centre slightly towards the down going blade.
As both engines and propellers rotate the same direction (clockwise when you're sitting in the plane looking forwards) the thrust from the left engine is slightly offset towards the centre of the plane, and the thrust from the right engine is offset slightly away from the centre of the plane.
In a situation with the left engine running and the right engine failed all the thrust is produced from a point inboard from the left propeller hub. This creates a moment around the centre of gravity and the plan will yaw. This is corrected with rudder.
In the opposite situation with the right engine running and the left engine failed all the thrust is produced from a point slightly outboard of the right propeller hub. This once again creates a moment arm and yaw, but this time the moment arm is larger because the point where thrust is produced is further away from the centre of the plane (centre of gravity). A larger moment means more yaw, and therefor more rudder required to counter it.
Diagram of Yawing Moments