Above is the default way tangents look. As you may see, the curves flow very evenly through the keys. Also, the handles are very tiny boxes on the end of the lines. These indicate Non-Weighted Tangents. To turn on the magic of Weighted Tangents,: Curves > Weighted Tangents.
Now all of the tangent handles have changed to solid spheres, the sign of Weighted Tangents. However, this is just an indication that the tangents now are weighted. If you try to adjust the curves, they still act the same as the Non-Weighted curves. To truly make the Weighted curves usable, there's one more thing to turn on.
The solid spheres on the ends indicate Weighted, but not Free tangents. This means that the actual weight, or influence, of the tangents is still locked at 50/50. in order to start really editing the curves, you need to free up the weights. There are two buttons on the toolbar concerning weighted tangents. One unlocks the weights so you can edit them, and the other locks them back into whatever place you've moved them into.
After unlocking the weights, the tangent handles become hollow squares. NOW you can fully utilize weighted tangents. When editing handles now, instead of only being able to move the handles up or down, you now
have the option of moving them horizontal as well. The handles now have a full range of motion.
Weighted tangents are the easiest tools to utilize when adding the animation principles of Smooth Ins and Outs to your project. Obviously, the point of using weighted tangents is to control how much influence a keyframe will have in an animation. This means controlling how fast or slow curves reach their max values. To make sure that a key frame is slowly eased into, you would pull out the left tangent handle to the left, toward the previous keyframe. This will make the curve's arc occur sooner and the character or prop's movement will snap out of the preview keyframe and smoothly roll into the current keyframe.
To ease out a pose or key, you would do the opposite. You would extend the right handle toward the following keyframe.
You can get some pretty crazy looking curves using weighted tangents, however, you can also use less keyframes and have very smooth animation curves.