"It's hard to talk about the Trinity," a Christian friend of mine once said with a sigh, "without falling into some heresy or other."
Patripassionism is what the Sabellian heresy was usually called in the Latin-speaking western half of the Roman Empire. The word "Sabellianism" is derived from the name of Sabellius, the priest who popularized the heresy in the second century. "Patripassionism," however, does not come from a personal name. Rather, it is a descriptive term that is derived from the Latin words pater, "father," and patior, "to suffer." (Before becoming a word for the unique sufferings endured by the lover, "passion" referred to suffering in general; hence "compassion," which simply means "suffering together.")
Patripassionism, then, is the doctrine of "father-suffering." Patripassionists believed that God himself suffered and died on the cross. Like all monarchians, patripassionists wanted to preserve the unity of God at all costs; they feared that they would slip into polytheism if they placed too much emphasis on the distinction between God the Father and God the Son. The interesting side effect of this theological stance was that it "demoted" God the Father to a changeable being, that is to say, a being that could feel emotions and experience pain.
Hippolytus of Rome blames one Noetus of Smyrna for creating this heresy, around the very beginning of the third century. We are told that Noetus taught "that Christ was the Father, and that the Father was born and suffered and died." Noetus' view does seem to develop logically from Trinitarian theology: if the three Persons of the Trinity are in fact one, then presumably they all die when one dies -- and no Christian denies that Jesus died, right?
But the orthodox church would eventually reject Noetus' position, and both he and his later followers, including Sabellius, were condemned. An elaborate system was put in the place of monarchianism, in which three hypostases ("essences") were said to exist in one ousia ("substance"). This allowed each person of the Trinity to be made of the same "stuff," while at the same time maintaining their separate realities. Basil of Caesarea tried to illustrate this concept with the example of a rainbow: a single thing with seven distinct essences. Whether the "green" of a rainbow suffers when "indigo" suffers is a question I am incapable of answering; I leave it as an exercise for the reader.
What is really at issue here is that Christians expect God to be unchangeable, while Jesus quite clearly underwent the same sorts of changes that affect all human beings. Patripassionism is difficult for orthodox Christianity to accept since suffering implies change and movement in time, while God the Father is believed to exist beyond time.
Note that "Patripassionism" can also be spelled "patripassianism".