What branch of science that deals about teaching or being a teacher?

I don't know that there is a specific branch of science dedicated to the study of teaching/teachers, but I did find some good resources that you might want to look at. Hopefully, they may answer your question, as well.

Degree programs that offer a master of science in science education degree:

http://www.Montana.edu/msse

http://www.icr.edu/se

http://www.umassonline.net/degrees/Online-Degree-Master-Education-Science-Teachers.cfm

A paper on the question from professor David Hestenes' point of view:

http://modeling.asu.edu/r&e/Wherefore_SciOfTeaching.PDF

An Annenberg Media site if you are interested in becoming a science teacher:

http://www.learner.org/workshops/science