I am not fantastic at explaining things but I'll give it a shot.
There are two ways of using the pen tool (bezier tool) in illustrator (or photoshop, In Design etc...) The first is to simply construct a path by point to point clicking, this method will not yield curves.
The second method involves dragging tangent handles:
The tangents you see when creating a curve do two things: Firstly they control the direction of the path, and secondly they influence the extent to which the path will keep that direction.
(It is difficult to explain this without getting too tech heavy...and I don't know enough about vectors at this time) create a graph In illustrator with both positive and negative x and y axes and number it. Place a point at 0,0 and drag the tangent handle to 5, 2. Now place a point at 5, 0 and drag a tangent to 7, 2.
What you should see is that you will generate a curve that roughly follows the direction of the furst tangent and forms a crest at 2.5, 0. From here the first tangent loses its influence as the path heads towards the second point, the change in direction is pretty constant at first but as the second tangent's influence grows stronger the path will suddenly drop into the negative y axis at 3.7, 0...form a crest at 4.2, -0.8 and rejoin the second point at 5, 0.
The reason for this shape is as follows, the first tangent is longer and has a greater control over the direction and magnitude of the curve (hence the name vector curves). Consequently the path travels pretty smoothly in this direction. At some point though, the path must yield to the second tangent you created. By dragging the tangent to 7, 2 what you have done is also created a tangent that hits 3, -2. It is this tangent that controls the second part of the curve. (this is something you'll see better by doing, as opposed to just reading :confused: ) So to follow it, the path that has been duefully following the first tangent must now follow the instructions of the second tangent. It does this at a much more advanced position along the line however because the first tangent is longer and stops at 5, 0. At some point however the second tangent must win and hence the path swoops down to pick it up before arriving at the second point. The other side of that handle will have an influence on the path if you choose to add a third point, and so on...
Gonna wrap things up shortly but just to throw one more thing in, if you had <alt> clicked on the second point and dragged, this would have created a cusp. Dragging the second tangent would not have any effect on the curve between point 1 and point 2.
Probably gonna get flamed for getting too technical (or for you mathematicians, too vague) but like I said, I'm not too good at explaining what I know :D ...just have a play
*edit* I just saw you posted a link...I didn't click on it and just waffled on... sorry :o