For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
Father Latour and Father Vaillant move from Sandusky, Ohio to the barren frontier of New Mexico. This area has only recently become part of the United States, and the two men are there to take control of the Catholic churches in the area. Once they get there, they make friends with some of the locals. But many of the people in New Mexico are stuck in their ways and aren't going to change just because the area belongs to the U.S. now. Latour and Vaillant have a lot of hard and annoying work ahead of them.
Act II
Fathers Latour and Vaillant travel around to many of the remote areas of their district and introduce themselves, saying that they are now in charge. Some of the Spanish-speaking priests welcome them humbly, but two in particular—Martinez and Lucero—create their own rogue churches instead of giving up power gracefully. Eventually, the two men die and the problem goes away. For Latour, though, that doesn't solve the problem of how he's going to get a bunch of Mexican Catholics and Native Americans to buy into his religious ideas.
Act III
Father Vaillant decides to leave his post at Father Latour's side to travel to Colorado. There has been a big gold strike there and thousands of men are moving to the area and engaging in lives of greed and alcoholism. Vaillant wants to do his part by bringing these men to the church. It's the right thing to do, but it leaves Latour without his closest friend. The novel speeds up drastically after this, fast-forwarding at least thirty years to when Father Vaillant has died of illness. Latour (who has made the rank of Archbishop) has built a giant cathedral in Santa Fe, and when he gets sick, he knows he'll be buried inside this church.
At the end of his life, Latour looks back on the work he has done in New Mexico and decides that overall, he's done a lot of good in the world. Then he slowly slips into unconsciousness and dies. As a sign of respect, people come from all over the area to pay their respects to Latour and to honor the work he did in and for New Mexico.