Modalism is the belief that God, rather than being three persons, is one person who reveals himself in three "modes," much as an actor might play three roles in a movie. It is also called Sabellianism or monarchianism.
The United Pentecostal Church and Apostolic Church are sister organizations. They are Pentecostal in the sense that they believe that Christians need a baptism in the Holy Spirit that is evidenced by speaking in tongues.
However, most Pentecostal groups consider the baptism in the Spirit a separate experience from salvation. The UPC and Apostolic Churches do not, with the result that they are forced to conclude that those who do not speak in tongues do not have the Holy Spirit and are not saved.
I have met several UPC members who are unwilling to hold such a stringent view, so it is not universal among them.
This belief is the equivalent of what is now called "Jesus only." It is held by very few denominations. The United Pentecostal Church and the Apostolic Church are the largest sects that hold to modalism.
Origins of Modalism
Modalism was the belief of two notable early church figures, Praxeas and Sabellius, both of whom aroused a large following in the church in the late 2nd (Praxeas) and early 3rd centuries (Sabellius). The size of their following and an explanation for it is given by Tertullian in A.D. 200:
The simple, indeed, (I will not call them unwise and unlearned), who always constitute the majority of believers, are startled at the dispensation [of the Trinity], on the ground that their very rule of faith withdraws them from the world's plurality of gods to the one only true God.
They fail to understand that, although he is the one only God, He must yet be believed in with his own order. The numerical order and distribution of the Trinity they assume to be a division of the Unity, whereas the Unity which derives the Trinity out of its own self is so far from being destroyed, that it is actually supported by it.
They are constantly throwing out against us that we are preachers of two gods and three gods, while they take to themselves the preeminent credit of being worshippers of the one God, as if the Unity itself with irrational deductions did not produce heresy, and the Trinity rationally considered constitute the truth. (Against Praxeas 3, emphasis mine)
Of the early Christian writings which remain to us, there is an astonishing consistency in their support for and explanation of the Trinity. Most elders in the Church were Trinitarian, though there were likely numerous modalists at the Council of Nicea, which condemned Arianism, a much different heresy.
Modalism in the Rest of Church History
No writings of Praxeas or Sabellius survive to today because they were considered heresy by the Church.
It is very likely that most modalists became satisfied with the Nicene Creed when it was redefined by the Athanasian Creed in the middle of the 4th century. It is thought that modalist bishops and Nicene bishops allied together against the Arians, who were still numerous after Nicea.