Waves are often represented like the example links above - something is varying "up and down" from the center line, and the distance between adjacent up peaks, down peaks, or any congruent parts of the wiggling line may be used to define the wave length.
This can be confusing because we are familiar with water waves where the surface does move up and down.
Your question seems to be, "What is wiggling?" in the sense of "What is actually moving?"
The representations show a variation in a value by graphing the value "up and down" on the graph... but that is the "mathematical" view. The vertical axis does not have to represent lateral motion of the thing - it might be density, or temperature, or color, or some other abstraction that has no physical correspondence to a gross movement.
So, the value wiggling up and down on the graph does not necessarily imply that there is a physical thing actually moving up or down or making an actual wiggling motion with respect to the center line - it is just a way of showing the changing value of something (something that does not have to really move physically to change value).
If I graphed how bright it is outside for a week I would get an overall wiggly line that had a wavelength of one day and there would be seven periods of that wavelength. You would not ask "What is wiggling?" in the sense of a cyclic motion back and forth about that when you knew that the measure was of brightness... right? The brightness changes value without moving... the crests and troughs have a vertical distance between them - that is how they are defined and located on the graph for measuring the wavelength between crests or between troughs, but that vertical component of an artifact of the graphical representation. In the brightness example, the brightness is certainly not "moving up and down" as a physical motion.