Roman cognomen meant "red-haired" in Latin. A few saints used to have this name including one Saint mentioned in one of Paul's epistles (New Testament.)
It was used by William II Rufus as a nickname. He was a king of England and used the name because of his red hair.
General usage is in the English speaking world.
I have a scotty/beagle mix named Rufus (his original name was Johnny, but I have a son named Jon and didn't want my kid and dog answering to the same name!) Ruf is a very friendly, outgoing, vocal dog, who is loved a great deal! Incidentally, the name Rufus means red-haired, even though my Rufus is a dark brindle.
"Rufus" is Latin for "Red" and hence it came to pass that my Irish Setter acquired the name. He was a very smart dog who figured out,on his own, how to go into the local grocery store, make a friend in the meat department and return home with a bone. Miss you "Ruuf".
I have a beagle named Rufus. I thought it was a cute name for a dog because a dog already says "Roof! Roof!" - short for Rufus.