The superscription of this Psalm 51, written centuries after the Psalm itself, connects it with the foul actions of King David in 2 Samuel 11-12, where the king commits adultery and then has the husband of the woman murdered, lying about both acts all the while. Such behaviors could indeed call forth deep penitence, and seeing the disaster that David's life becomes after these events, one may only hope that the king resorted to such penitence for the sake of his own soul. Knowing David as the text allows us to know him suggests, unfortunately, that penitence seemed low on the king's desired activities.
Let me suggest two things to note about this marvelous poem. First: do not allow the lectionary collectors to stop you at verse 12. After the penitent has begged God for succor in the face of his/her deepest sins (Ps. 51:4) and has expressed the hope of being forgiven by that God (Ps. 51:6-12), the poet then promises God that she/he will become an evangelist for the forgiving God (Ps. 51:13), and will avoid the actions of revenge against the enemies (Ps. 51:14), an action demanded in more than a few other psalms (see 137 and 58 as especially crude examples).
Second: Psalm 51:15-17 is a magnificent announcement of what our God finally wants from us as God's people. When we "open our lips" we vow that we will "declare your praise," as opposed to cursing our enemies or getting all the goodies that we can possibly hold. Furthermore, God cares not a fig for burnt offerings, a claim that no ancient Israelite priest would welcome, though such an idea is found throughout the prophets (see Amos 5 as the quintessential example). The sacrifice God wants, it turns out is a "shattered spirit," a "shattered and contrite heart" (Ps. 51:17).
What can that possibly mean in 2015? Going to church is not automatically pleasing to this God, or service on numerous committees, or generous gifts of one's money. These are nice things, but hardly what God wants in the end. I would suggest that "shattered and contrite" hearts (the seat of will and intelligence for the Hebrews) and spirits (the word also means "wind," and is here the animating force of a person —see Ezek. 37 for a long story illustrating that reality) mean that what we most lack in our time is humility. We believe too fervently in our own competence, our own brains, our own plans, to the exclusion and rejection of the competence and brains and plans of those who think differently than do we. The inability of our governments to agree on much of anything is a direct result of a lack of humility, the belief that any idea that is not mine might have merit. In the religious life it means that those who do not believe as I believe are wrong and in extreme cases, too many of which we see each day, must be eliminated.
Give us, O God, shattered spirits this day so that we may open our eyes wide enough to see the value in others of your vast created people. Now that, I dare say, will preach!