On a touch device (tablet or smartphone for example), is it valid to say that the buttons must be round, because the end of my finger is round? More specifically, will the user understand more easily that an element is clickable if this element is circular?
Nb : as a non-native English speaker, I don't know if I have to say "round" or "circular" or any other word. What I mean is "the shape of a circle".
No, not necessarily. In the past, designers used heavily skeuomorphic patterns -- visual styles that strongly resembled physical, tangible objects in order to suggest how they can be interacted with. Physical buttons are often round or shaped to someone's finger because the user is physically touching them, and early digital designers wanted to make sure users made the transition from analog to digital more easily.
While skeuomorphic designs may have been helpful when people were just learning about computers, increased familiarity with digital interfaces has (I think) decreased the need to focus these kinds of affordances. It's still a good idea to make buttons appear interactive (e.g. with a subtle gradient, or appearing 3D), but I don't think there's really a need to make them round for someone to make the connection to their finger. Plenty of successful interfaces use completely squared elements (e.g. Windows 8 tiles) or icons.
Interesting is the general form of the question: Does a user understand how something is intended to be used because it's shape is an ergonomic match for a body part?
Examples
Buttons have been culturally well understood for many generations which has let them evolve away from a round shape. e.g. audio cassette players in the 70's with their rectangular buttons.
The article here covers the best practices for touch targets. The rules are not so much based on the shape, but size.
No, definitely not.
Rounded edges are just another piece of the aesthetic formula as you build your UX. I've found that playing with 0% rounded edges allows for an almost fun, "flat design" aesthetic so popular in 2014.
It seems to me button shapes are more about the connection to the functionality, than it is about the connection to the finger. For example:
Well... are buttons round everywhere in real life?
Look at your keyboard. How many 'round' buttons are there?
No. I cannot speak for the Apple ecosystem, but if you look at Google's Android, or Windows for the past ten years or so, even Linux, circular shapes for buttons can be quite rare.
I would assume that rectangles are ideal because width implies that something can be swiped horizontally.
If you look at older touch screen devices and machines, those designed for stylus or resistive input, the traditional desktop metaphor is often maintained, with no touch friendly enhancements.
