For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
Bruce Lyon
The question is: "What does it mean that God will not despise a broken spirit and contrite heart?"
Psalm 51:17 says, “My sacrifice, O God, is a broken spirit; a broken and contrite heart you, God, will not despise.” The meaning of this is connected with the verse just before it. Verse 16 says, “You do not delight in sacrifice, or I would bring it; you do not take pleasure in burnt offerings.” David is stating that there is nothing we can offer God to appease Him when we have sinned. More animal sacrifices were not what God was looking for. GOD DESIRES TRUE REPENTANCE FROM US.
David reminds us that the only path to forgiveness is a broken heart and a humble spirit (Matthew 5:3). When we throw ourselves on the mercy of God, He delights to lift us up (Luke 18:13-14). When we openly acknowledge our sin against God, turn from it, and cry out for cleansing, God promises that He will hear us and forgive (1 John 1:9).
It is interesting to note that, although David sinned against Bathsheba and her husband, he makes this statement to God: “Against you, you only, have I sinned and done what is evil in your sight” (verse 4). David gets to the heart of why God so hates sin. It is a violation of His very nature. We are created in that image, but our sin mars it, like a smudge on a mirror. A broken spirit and a contrite heart invite God to clean that smudge and restore us to right relationship with Him.
These words are not mine but from http://www.gotquestions.org/. Whoever wrote this said it better than I could have done and answers the question "What does it mean that God will not despise a broken spirit and contrite heart?"
Suzan Toureau
God won't hold our sin against us if we turn away from it and confess honestly to him.
Bruce Lyon
A contrite attitude of mind comes from a person who has come to realize their poverty of spirit, their total incapability to do anything that is pleasing to God of and by their own self will, impossible. Realizing that they are poor in spirit and needing God's Spirit indwelling them to accomplish all that He requires of them as His adopted sons/daughters, they indeed come before Him in an attitude of repentance and asking for His forgiveness.
Now, God responds forgiving that son/daughter because of the sin offering sacrifice of His beloved son Jesus who by his sacrifice, his shed blood has reconciled all mankind unto his God and his Father.
We should also realize that we are bought and paid for by the blood of the lord Jesus and are no longer our own. We are his slaves and as such have no right to act according to our self-will anymore. We now become as Paul says slaves of righteousness - right doing in faith obedience to the will of God and His son Jesus our lord!
The amazing result of our becoming a slave of the lord Jesus is; he as our master, our slave owner is required to provide for us in every way and enable us to live according to the righteous requirements of his God and his Father and ours! He came to indwell all those whom his God and Father will call, by through the power of His Spirit which He has given to His son Jesus, to give unto those that are His. It was Jesus who poured out the promised Spirit of God at Pentecost!