Speed graph handles are always horizontal. The length of the yellow handles determines the curve of the speed graph. The graph represents acceleration. In your example, the movement of the layer (the position) accelerates at a fairly constant rate until you get to about 12 seconds, then the acceleration increases until you get to about 18 seconds, then decelerates quickly until it abruptly stops at the last keyframe.
Think of it like you were driving a car. You are stopped, then you press gently on the accelerator as you move away from the intersection. it takes about 12 seconds to cross the intersection. The road ahead is clear, so you press the gas pedal farther and farther down as you accelerate toward the speed limit. Twelve seconds into your trip, someone pulls out of a side street, blocking the street. You let off the gas and move your foot to the brake. That gives you the little hump at the top of the curve, then you press hard on the brake to try and stop before you hit the car. Unfortunately, just before you come to a smooth stop, you hit the other car resulting in a sudden stop. That is what your speed graph is telling me.
Value graph handles are bezier handles. They are also straight, but you can place them at any angle. That changes the path of the car. It is extremely difficult to manipulate a path using the graph editor, so the speed graph is the primary tool for adjusting when something happens. If you need to adjust the path, you can use the pen tool on the Motion Path in the Composition panel.
This tutorial I created should clear things up.
I hope this helps.
My After Effects Quick Tips
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