This great psalm of repentance recounts David's personal moral history, and in these translations, it goes all the way back to the moment of conception! An unviable tissue mass or a blob of protoplasm is not—indeed cannot be—a moral agent. These translations indicate a moral disposition of a moral agent at conception!
John W. Ritenbaugh
When David saw the enormity of his sin, he realized he had hurt God and His purpose. His sorrow, chagrin, and remorse reached deeply into his heart, mind, and entire being. Our opposition to God should create a similar deep emotional response in us, for we have all played major roles in our Savior's death. He died for our sins. Emotional sorrow alone is not the answer, however. Paul says godly sorrow produces repentance (change) toward salvation, while worldly sorrow is like saying, "I'm sorry I got caught. I'll be more careful next time I sin."
Martin G. Collins
Basic Doctrines: Repentance
Those who believe in original sin believe that David is saying that he was born a sinful person. However, that is a misinterpretation.
The truth is a bit simpler. He means that he was born into a sinful world, and even his mother, whom he probably loved more than anyone else other than God, was a sinner. Put in other words, his mother was a sinner when he was conceived, and when he was born, the world was full of sin. Sin and sinful people had surrounded him his whole life. A baby is born a fairly blank slate, certainly having done no sin, but the influence of sinners along with his fleshly nature invariably get the better of him, and before long, he sins. In his way, David is telling God, “With all the sin around me, it's not surprising that I sinned too.”
In the next verse, he declares that God “desires truth in the inward parts” (Psalm 51:6). In other words, He wants people to change their natures, which only He can help us to do, as the last half of the verse explains: “And in the hidden part You will make me to know wisdom.”
David is essentially repeating what God said in Genesis 8:21, ". . . man's heart is evil from his youth." But he wanted his nature changed from evil to good, from dirty to clean, from lies to truth. Only repentance and a renewed relationship with God would enable that to occur. And, truth be told, it will never be completely changed until the resurrection from the dead.
Richard T. Ritenbaugh
Are Humans Good or Evil?