Is there anything I can do if a professor agreed to supervise me but I was rejected by the admission committee? Some time ago I contacted a professor at another university and, after some discussions, he agreed to supervise me and told me to make a formal application to the department. Unfortunately, the admission committee rejected my application. I asked that professor why, and he said he had no idea and I don't know if he tried to follow my application. 

I am sure I meet the program's minimum requirements, submitted all necessary documents and even indicated that professor as a supervisor in a dedicated field. I would like to ask if I can do something, like making an appeal (although the department has no such thing), or keeping on sending email to the professor? Is this situation common?
No PhD program that I am aware of guarantees admission to every candidate who meets the minimum requirements for admission. Moreover, an agreement to supervise you is not a guarantee of admission, either.

Most likely what happened is the usual in such circumstances—there were enough other qualified candidates that the admissions committee did not choose to extend you an offer. I'm afraid that there's not really much you can do here. Writing the professor will be a waste of your time, and filing an appeal isn't likely to get you anywhere, either.