Psalm 51 is a great Lenten Psalm, which is why we say it every year at the Ash Wednesday service and, in Year B of the Lectionary, for the 5th Sunday in Lent. It is particular appropriate because of its history.
Psalm 51 was written by David in a time where he needed to repent and return to the Lord. What really happened for David was a sin compounding on a sin.
David, as king, should have been with his troops in battle. Instead he was at home in his palace. This is actually something the Prophet Samuel warned the Israelites would happen if they adopted kings as rulers.
Because David stayed, he caught sight of Bathsheba, leading to a moment that in modern times we could only describe with "#metoo". (You can read more about David's wrongdoing by clicking here.)
David's sin was so great that the Prophet Nathan came and told him off for what he had done. David, to his credit, and after an illustration from Nathan, immediately repents.
Psalm 51 gets at what David was feeling at this time. It is a good Psalm for us in times when we need to repent too.