Is the order of authors' names important?
The answer depends on the field of research. For example, in most areas of mathematics and some of theoretical physics, authors are listed alphabetically. In other fields this is different, and there typically the first author is the one that did most of the work. In such fields the first author position is the most prestigious one. The last author typically has a role more on the background, for example by posing the research project and/or playing an advising role.

I have never heard of any convetion where the supervisor is the first author if they didn't do the majority of the work.