Mormons believe in eternal progression, though we aren't always sure what that means. Joseph Smith didn't use the phrase, though Brigham Young did. The notion seems to have been founded primarily on teachings late in Joseph Smith's life, just before his martyrdom in 1844.
The primary textual evidence we have for the teaching comes from a sermon that Smith delivered in April of that year at a funeral. That sermon was not recorded stenographically. Instead, it was reconstructed from the notes of several in the audience. People of the time were more accustomed to remembering the content of sermons than we, and those who wrote down their recollections did so soon after they heard Smith preach. Nevertheless, we have to remember that the sermon as we have it today is a reconstruction rather than a transcription. That said, there is no question that Joseph Smith taught some surprising things in it.
Responding to the death of a man named King Follett, Smith said that he wanted to talk about the loss of loved ones in general terms. To do so he began by speaking about "the character of God," saying, "If men do not comprehend the character of God, they do not comprehend themselves." Knowing that and the teachings related to it, Smith said, "are the first principles of consolation."
He went on to say, "We have imagined and supposed that God was God from all eternity. . . . [But] he was once a man like us; yea, . . . God himself, the Father of us all, dwelt on an earth, the same as Jesus Christ himself did." The Prophet added:
Here, then, is eternal life—to know the only wise and true God; and you have got to learn how to be gods yourselves, and to be kings and priests to God . . . by going from one small degree to another, and from a small capacity to a great one; from grace to grace, from exaltation to exaltation, until you attain to the resurrection of the dead.
Presumably Brigham Young's phrase, "eternal progression," has its origin in Smith's description of the necessity of going from one small degree to another.
Not all Mormons believe that God was once a human person—it is not a doctrinally binding belief—but many do. From Parley and Orson Pratt in the 19th century to Blake T. Ostler in the 21st, Mormon thinkers have sought ways to explain that unusual claim. More of us, however, believe in eternal progression, even if we aren't sure what we mean by the term.
For me the idea that God was once human is literally unthinkable: I don't understand it or know how to think about it. But that may only reveal my inability. Whatever the case with that belief, I'm less interested in the question of whether God was once a human being than I am in what the notion of eternal progression might mean for we who are clearly human.
One of Joseph Smith's canonized revelations says, "he that receiveth me receiveth my Father; and he that receiveth my Father receiveth my Father's kingdom; therefore all that my Father hath shall be given unto him" (D&C 84:37-38). Clearly that is a repetition of the teaching in Romans 8:
For as many as are led by the Spirit of God, they are the sons of God. For ye have not received the spirit of bondage again to fear; but ye have received the Spirit of adoption, whereby we cry, Abba, Father. The Spirit itself beareth witness with our spirit, that we are the children of God: and if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together (Rom. 8:14-17).
To be in Christ is to be a member of his family, and to be in his family is to be a child of the Father, an equal inheritor with Christ. Paul's promise is that we can go from whatever we are to that.
The scriptures—Old Testament, New Testament, Book of Mormon, Doctrine and Covenants, Pearl of Great Price—are aspirational. Recognizing our humanity, they teach us that we can be more, and the "more" that we can be is an eternal more, qualitatively eternal now in the hope of post-mortal eternity. Scripture is realistic about who we are and idealistic about who we can be.
Modern culture asks us to recognize our humanity, to accept our weaknesses, to avoid the neuroses that perfectionism can bring. The trick for Jews and Christians is to avoid those traps without giving up the aspirations of scripture: "Ye shall be holy, for I the Lord your God am holy" (Lev. 19:2) and "Be ye therefore perfect, even as your Father which is in heaven is perfect" (Mt. 5:48). Whatever else it may or may not mean, eternal progression is a Mormon way of talking about the aspiration inculcated by those commandments: be holy; be perfect.