What does Psalm 51:19 mean?

In this verse David anticipates the Lord's acceptance of His people's sacrifices when their hearts are right with Him. He mentions burnt offerings and whole burnt offerings.

Leviticus 1:3 specifies that the worshiper who offers a burnt offering must offer a male without blemish, and "he shall bring it to the entrance of the tent of meeting, that he may be accepted before the LORD." Leviticus 1:4 instructs, "He shall lay his hand on the head of the burnt offering, and it shall be accepted for him to make atonement for him." When a bull was sacrificed as a whole burnt offering, the worshiper was instructed to lay his hand on its head. The blood of the bull was sprinkled by the priest seven times before the Lord in front of the sanctuary's veil and then poured out at the base of the altar of burnt offering (Leviticus 4:5–7). The bull's fat covering its entrails, its kidneys with the fat, and its liver were burned on the altar, and then the bull's skin, flesh, head, legs, entrails, and dung were burned outside the camp.

The worshiper's identification with the sacrifice by laying his hand on the animal's head prefigures our identification with Christ as the perfect sacrifice for our sin.