From Matthew Henry's Concise Commentary
The psalmist prays for mercy, humbly confessing and lamenting his sins. (1-6)
He pleads for pardon, that he may promote the glory of God and the conversion of sinners. (7-15)
God is pleased with a contrite heart and a prayer for the prosperity of Zion. (16-19)
Note that the first verse (Psalm 51:1) is actually the preamble. The next verses (Psalm 51:1-2) is where the psalm begins. I set it up this way for ease of memorizing.