the term co-emperor comes from the word "symbasileus", as it is used in-game, which was often, though not always, the title given to the man chosen to inherit the empire as the next emperor. if a father wanted his son to rule after him, he would raise his son to the level of co-emperor in order to prepare him for the monumental task of governing an empire. also, by bestowing the title, it was clear to the nobility & senators who the next emperor would be. don't be confused, though; despite it translating to co-emperor, it does not mean they were equals in terms of power. The emperor was ruler. It's very similar to what Augustus did for Tiberius in the latter years of his life; giving his heir the positions/powers that he himself had occupied/used as Princeps, thus marking him as the next Princeps.
regents are rulers outside of the established blood line.
but yeah, in terms of game mechanics, it's the same as every other faction. it's based upon whether or not the leader has a son of-age, if not it goes to whoever has the most authority.