Psalm 51:18-19 meaning

Having petitioned God to forgive him and renew fellowship with him, David now turns outward, to his kingdom and the people God gave him to steward:

By your favor do good to Zion;
Build the walls of Jerusalem (v 18).

By the time of David, the terms Zion and Jerusalem were often used interchangeably. Jerusalem was Israel's capital city, in which David dwelt. And Jerusalem was built on a number of hills, including Mount Zion. 

Additionally, Zion and Jerusalem could refer to the entire kingdom of Israel. They were kingdom words that evoked kingdom images. Just as David petitioned God to rebuild his own inner being through a new, clean heart, he now asks God for a similar foundation for the rebuilding of the nation and people of God. 

David asserted in verse 16, "For You do not delight in sacrifice"—rather God delights in a contrite heart that will follow His ways—the ways that lead to life rather than destruction. Now David says, 

Then You will delight in
righteous sacrifices,
In burnt offering and whole
burnt offering;
Then young bulls will be
offered on Your altar (v 19).

The reason God now will delight in righteous sacrifices is because they are external images of an internal reality of submission to God and His ways. This would indicate that the physical image in the prior verse reflects a deeper spiritual reality. David says:

By your favor do good to Zion;
Build the walls of Jerusalem (v 18).

Apparently by this he is speaking of God's favor to do good to Zion as moving in the hearts of the people, and renewing their spirit. The physical image of the walls of Jerusalem is an image of strength. These were walls that already existed, and were built by humans. It seems here that David is asking God to provide another kind of protection, a protection much stronger than that of a mere wall—David desired God to build the spiritual vitality of Israel. 

In the context of verse 1 when he appealed to the "gracious" "lovingkindness" of God, David appeals to God's favor when he says By Your favor do good to Zion. "God's Grace" in scripture is simply God's favor. There is no standard apart from God by which anything can be measured. All of God's judgements are just. Therefore, no favor of God is ever owed by Him. It stands then that all favor by God is a matter of grace. 

It seems to follow that when David asks God to Build the walls of Jerusalem, then his nation will now offer ritual sacrifices from a clean, pure heart. Young bulls will be offered on Your altar likely is a description to a burnt offering or whole burnt offering. These offerings were prescribed in the Mosaic Law, and were to be offered at prescribed times (Deuteronomy 12:11). In many cases the animal sacrifices were to be offered then enjoyed by being consumed in a celebratory festival. The inner restoration of fellowship between us and God ought then to flow into our relationships with others, and create fellowship within our communities. 

God's delight is no small thing to attain. God's delight is to bring "lovingkindness, justice and righteousness on earth" (Jeremiah 9:24). When God's people reflect His righteousness in their lives, when they are dedicated to "do justice, to love kindness, and to walk humbly with...God" (Micah 6:8) then their religious practices—their righteous sacrifices and burnt offerings—resound with an authenticity that lifts others and greatly pleases God.  Temple rituals—mere shadows of a greater redemptive reality (Hebrews 10:1-4)—were no longer useful following the death and resurrection of Jesus, God's final and complete offering of sacrifice for human sin (Hebrews 9:11-14).

An application of this psalm is that external religious observance is of no interest to God unless it is an external application of a pure heart that is dedicated to following His ways (Psalm 51:16-17). As we can see in the context for this psalm, by breaking God's law, David committed murder, causing unnecessary death, and the breakup/death of a family. God's law sets forth a way of living that reconnects people with God's (good) design for the world. Following His ways brings light and life. Our ability to follow God's path requires that we approach Him with a repentant heart when we fail, and receive forgiveness and a restoration of fellowship. 

This principle is also reflected in the "Model Prayer" also called the Lord's Prayer. The central point of that prayer is that fellowship with God requires that we forgive others as we desire to be forgiven by God. Our fellowship with God results in fellowship with one another (1 John 1:2-4). 

Finally, for the believer, in the spirit of Psalm 51, we are to regard ourselves as "living sacrifices" (Romans 12:1). That is, our entire lives are to be continually offered as a sacrifice to God and His perfect will for our lives. We are to sacrifice "daily" by taking up our cross and following Jesus's example (Luke 9:23) 

If we do not offer our life as a sacrifice we will surely lose it—we will not live in harmony with God, but suffer the joyless and dreadful consequences of doing life on our own (Luke 9:24a). But if we offer our lives as a sacrifice for His sake, we will surely find it (Luke 9:24b)—and experience the joy that David found in God through his confessed brokenness, the same joy Jesus attained when He cared nothing for the shame of the cross and instead viewed only the reward set before Him in obeying His Father (Hebrews 12:1-2).

Psalm 51 is a prayer and model for the believer who has sinned to be restored to fellowship and harmony with God and His perfect will for our lives.