Numbers 19 gives the law of the red heifer. In summary, it says that the priest shall sacrifice an unblemished red heifer that has never been yoked. The heifer was to be burned in his presence, and cedar, hyssop, and scarlet thread were to be thrown on the fire once it was burning. The ashes were to be gathered and carefully stored as "a purification for sin." When any person touched a dead body, animal or human, they were unclean for seven days. On the third and seventh days they were to present themselves for sprinkling. A clean person would fill a container from a source of running water, then add ashes from the red heifer. He would use hyssop to sprinkle the unclean person. If this was done on the third and seventh days, and the unclean person immersed himself, then he became clean. The part that had stumped many scholars, supposedly including Solomon, is that the clean person who sprinkled the water and ashes then became unclean until evening. The clean person who sprinkles the waters becomes unclean, while the one who is sprinkled with the waters becomes clean.
The person who was sprinkled did not merely become ritually clean. God considered his flesh to be completely and effectively clean. That impurity would be remembered no more. It was not even to be remembered at the Day of Atonement, because it was totally gone.
Since most Christians today don't need to worry about the laws of ritual purity that the Jews had to concern themselves with, many might ask what significance the red heifer has to us. So it brought purity to the impure, and impurity to the pure. So what?
Actually, the law of the red heifer had more to do with us as Christians than most people realize. It has everything to do with our salvation. I know, we are saved by the faith of Christ. The blood of the sacrificed Son of God cleanses us, not the ashes of a red heifer. Still, the idea presented in Deuteronomy 19 is central to that cleansing.
In science it is the law of conservation of energy. Energy may be transferred, but it is neither created nor destroyed. In relation to sin, the law of the red heifer says that sin may be transferred but not destroyed. Why is that important to us? Because it explains why Jesus, the Messiah, died on the cross. It says that my sin had to be transferred to a clean person or else I would be "cut off" from God's people. Since there was no other person without sin (Heb 4:15), the only one who could effect the transfer was Jesus.
Jesus is our red heifer. More correctly, he is the clean person who sprinkled with the ashes of the red heifer. He became unclean for us. He took our sin on himself, that we might be pure. The ashes of the red heifer purified the flesh; he purifies our spirit.
For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh: How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God? (Heb 9:13-14)