What is the difference between the terms wave, waveform, and signal?
A signal is simply something observable changing across one or multiple dimensions. As that, it carries some changing state, which means it has some information¹. 

For example, air pressure changing over time might be an audio signal, brightness changing over length might be a barcode (signal), and voltage changing over time is what we typically call an electric signal.

Now, a wave is usually a physical entity that actually fulfills a periodic, harmonic motion. As that, it is a special kind of signal.

Waveform is a term from the radio (and possibly audio) engineering. In that, you modify a (usually harmonic), periodic signal (i.e. a wave) with varying parameters (i.e. another signal) to give it the form you want. Radio modulations are a typical example of that – FM broadcasting is a classical waveform to modify radio waves according to another signal (in that case, audio that you want to broadcast).

I'd strongly recommend using waveform only in a context where the usage of "I know there's information in this wave's modification over time" is important, wave only when you have a (typically propagating) periodic signal, and signal whenever you describe something changing in general.

People that do a lot of signal processing might actually imply periodic nature if they read wave. Things like "square wave" already feel a bit off, because they're not harmonic (within finite bandwidth).


¹I'll not go into the information theory aspects of that, because it will quickly expand far beyond the scope of this question, and will lead to me explaining a lot of stochastics.