It depends on the plane. Some planes will remain fairly level, and some planes will balloon up a little bit and climb when you apply the flaps. It can also depend on other things such as the speed of the plane when you apply the flaps as well. However, it's is extremely unlikely to nose down the plane. To understand this you have to understand the aerodynamic effect of flaps, and what they do when deployed. The curvature of the top of a wing causes lift as air flows over the curve (very simplistic explanation, but you get the idea). When flaps are dropped it has the effect  of extending the back of the wing and increasing the surface of the curvature on the top of the wing. Because of this the wing can produce the same amount, or more lift at any given airspeed. This will allow the plane to continue flying at slower speeds, which can allow it to land at slower speeds (one of the two times flaps are used). The other time flaps are used is on takeoff. The effect of the flaps increased the lift on the wing which allows it to get off the ground quicker, especially when it's heavier at takeoff. This effect is why most (but not all) planes usually balloon up when you apply flaps. When you drop the flaps the wings is now producing more lift at the same speed, which caused it to rise up.

Ken