To get a little technical on you:
When an engine fails (or has to be shut down for a surge, fire, etc), two things happen to the aircraft:
1. Only half the power is available - that means, the pilot will lower the nose, as to not loose speed. It is like driving up a hill in your car, and the car gets slower and slower, even though you are giving it full throttle. What can you do? In a car - not much, but in an airplane, by lowering the nose, we can adjust the "steepness of the hill", that means it won't climb as much, but you can maintain the speed (you need the speed to fly). That makes for the "sinking" feeling, mentioned earlier.
2. The remaining engine will pull the plane to the opposite side.
Both engines working together give an even pull on the plane, but only one left will pull it sideways, just as if you are held by both arms leaning backwards, and then one is let loose. You will not fall, but turn a bit, until stabilised on the one still holding you. Same goes for the plane, it will sway around a bit, until the pilot has manipulated the controls to find a new equilibrium for the aircraft. That way the airplane can be flown and landed, if not comfortably, though safely.
Nic