What is the role and function of the priest and the lieutenant in The Power and the Glory?

The priest and the lieutenant each represent opposing threads of a narrative argument.

The conflict between the novel's two main characters, the "whisky priest" and the lieutenant, parallels the tension between the novel's two overlapping themes—the spiritual theme of religious faith and devotion, and the political theme of the Church's obligation to aid the poor.

The priest can be seen to symbolize the church in its pride and privilege, its hubris and its decadence. However, despite his personal failures and even his hypocrisy, the priest is a sympathetic figure and a true believer. 

"When you visualized a man or woman carefully, you could always begin to feel pity — that was a quality God's image carried with it. When you saw the lines at the corners of the eyes, the shape of the mouth, how the hair grew, it was impossible to hate. Hate was just a failure of the imagination."

In thoughts of this sort, the priest shows his tendency toward abject humility and universal empathy, qualities which do not negate his failures but which do suggest some virtue in his character.

The lieutenant, in contrast, is a representative of the state. He is also a zealous man, dedicated to his own truth and to a sense of service. He is generous with the poor while also being rather vicious and violent. He wants to do good, but he is willing to kill innocent people to purge the state of religion. 

While the priest is morally compromised, he is associated with the "glory" of the novel's title. (This relationship may be somewhat ironic.) The lieutenant is associated with the "power" of the novel's title.