When in the context of theology, what does it mean to be righteous? How is this different from being holy?

The concept of righteousness was introduced in the very beginning in the Garden of Eden. The basis of righteousness is founded on the moral character of God and His standard of good and evil. It is precisely because of the absoluteness of God's moral standard that Satan desires to debase it with his temptation of Eve:

The serpent said to the woman, "You surely will not die! For God knows that in the day you eat from it your eyes will be opened, and you will be like God, knowing good and evil." When the woman saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was desirable to make one wise, she took from its fruit and ate; and she gave also to her husband with her, and he ate. (Gen 3:4-6)

When each person has their own personal standard for good and evil, relative moral standards becomes the norm.

After Adam and Eve disobey God's prohibition of eating fruit, God judges in a sequential manner: 1) Satan, the deceiving tempter, is cursed, 2) Eve is given two sanctions, and 3) Adam, for listening to the voice of his wife, receives a life of toil, because the earth is cursed (Gen 3:14-19). And they are expelled from the presence of the Tree of Life:

Then the Lord God said, "Behold, the man has become like one of Us, knowing good and evil; and now, he might stretch out his hand, and take also from the tree of life, and eat, and live forever — therefore the Lord God sent him out from the garden of Eden, to cultivate the ground from which he was taken. So He drove the man out; and at the east of the garden of Eden He stationed the cherubim and the flaming sword which turned every direction to guard the way to the tree of life. (Gen 3:22-24)

It is not until one reads the book of Revelation that the significance of the Tree of Life is realized. After the end of human history, the Tree of Life is located in the holy city of the New Jerusalem on the New Earth (Rev 22:1-5) where Believers are in the presence of God; it is only the righteous who have the right to the tree (Rev 22:18-19).

From broken relationship to restoration, God's righteousness is the standard by which God interacts with human beings and defines who His people are.

For human beings, righteousness is behavior that makes one acceptable to God. A relationship with God requires the conformation to His standard and fulfilling rightful obligations.

These are the records of the generations of Noah. Noah was a righteous man, blameless in his time; Noah walked with God. (Gen 6:9)

Then he believed in the Lord; and He reckoned it to him as righteousness. (Gen 15:6)

"For I have chosen him, so that he may command his children and his household after him to keep the way of the Lord by doing righteousness and justice, so that the Lord may bring upon Abraham what He has spoken about him." (Gen 18:19)

With the conditional Mosaic Covenant in view, righteousness is the type of conduct that maintains or restores one's relationship with God.

You should diligently keep the commandments of the Lord your God, and His testimonies and His statutes which He has commanded you. You shall do what is right and good in the sight of the Lord, that it may be well with you and that you may go in and possess the good land which the Lord swore to give your fathers, by driving out all your enemies from before you, as the Lord has spoken. (Deut 6:17-19)

"Doing right" meant obeying and following God's word as stipulated in the Ten Commandments which summed up the Mosaic Law. "Right" behavior was living in accordance to the Law, and righteousness was determined by whether one obeyed and followed God's word.

God's word, an objective standard which is outside of and an authority above a human being, is the means by which a person is evaluated for righteousness. It is not a subjective standard in which God chooses who is righteous or not.

When Jesus Christ arrives, He emphasizes the importance of pursuing the righteousness of God and its objective standard of determining who is good or evil and approved by God. Jesus Christ does not present righteousness as ethical perfection; but rather, as obedience to God the Father.

Then Jesus arrived from Galilee at the Jordan coming to John, to be baptized by him. But John tried to prevent Him, saying, "I have need to be baptized by You, and do You come to me?" But Jesus answering said to him, "Permit it at this time; for in this way it is fitting for us to fulfill all righteousness." Then he permitted Him. After being baptized, Jesus came up immediately from the water; and behold, the heavens were opened, and he saw the Spirit of God descending as a dove and lighting on Him, and behold, a voice out of the heavens said, "This is My beloved Son, in whom I am well-pleased." (Matt 3:13-17)

But seek first His kingdom and His righteousness, and all these things will be added to you. (Matt 6:33)

"Beware of practicing your righteousness before men to be noticed by them; otherwise you have no reward with your Father who is in heaven. (Matt 6:1)

Judaism viewed God's Law to include rabbinic tradition (i.e. Halakhoth), which were religious practices based on "the way their fathers walked and that which their children are bound to follow." For Jews, righteousness was conformity to the Law and these legalized customs with a focus on the attainment of reward. "Righteous acts" considered particularly meritorious were charity (i.e. material expenditure, etc.) and mercy (i.e. empathy with mourners, sick, etc.) both of which were often intentional overt public displays.

The term "holy" makes its first appearance when Moses encounters God for the very first time (Ex 3:5). Through the Mosaic legislation that takes place after the Exodus, God teaches the nation of Israel the meaning of "holy." It is a term that places an emphasis on God, because it calls to attention the condition something must be in when encountering the realm of God. To be holy, something must be set apart or separated from the profane or unclean.

The imperative "be holy for I am holy" has a special significance (Lev 11:44-45; 19:2; 1 Pet 1:13-25), because it comes after God reveals that the nation of Israel will be His own possession and intended to be devoted to Himself:

Now then, if you will indeed obey My voice and keep My covenant, then you shall be My own possession among all the peoples, for all the earth is Mine; and you shall be to Me a kingdom of priests and a holy nation.' These are the words that you shall speak to the sons of Israel." (Ex 19:5-6)

Thus, the nation of Israel was obligated to be holy. In making the conditional Mosaic Covenant, the nation of Israel learned what sin was and the concept of being holy. A human being achieved the state of holiness when one did "what is right and good in the sight of the Lord."

When God makes the New Covenant, He does so to vindicate the holiness of His great name, which the nation of Israel profaned among the nations they went to (Ezek 36:22-23). Defined by the presence of the Holy Spirit (Ezek 36:22-32), holiness under the New Covenant retains its Old Testament meaning of setting apart from the profane as a basis of belonging to God. When Jesus Christ arrives to inaugurate the New Covenant, it is through Jesus Christ that the Holy Spirit is bestowed that enables a Believer to be sanctified and made holy; it is the indwelling of the Holy Spirit that guarantees one's entrance into the kingdom of God and inheritance as God's adopted (Eph 1:13-14; 2 Cor 1:21-22; 5:5).

Jesus Christ is the manifestation of the righteousness of God, and it is through faith in Jesus Christ that one is deemed with this righteousness of God.

But now apart from the Law the righteousness of God has been manifested, being witnessed by the Law and the Prophets, even the righteousness of God through faith in Jesus Christ for all those who believe; for there is no distinction; for all have sinned and fall short of the glory of God, being justified as a gift by His grace through the redemption which is in Christ Jesus; whom God displayed publicly as a propitiation in His blood through faith. This was to demonstrate His righteousness, because in the forbearance of God He passed over the sins previously committed; for the demonstration, I say, of His righteousness at the present time, so that He would be just and the justifier of the one who has faith in Jesus. (Rom 3:21-26)

For we through the Spirit, by faith, are waiting for the hope of righteousness. For in Christ Jesus neither circumcision nor uncircumcision means anything, but faith working through love. (Gal 5:5-6)

Righteousness defines who is holy, set apart as God's people, and entitled to the promise of eternal life.

For as through the one man's disobedience the many were made sinners, even so through the obedience of the One the many will be made righteous. The Law came in so that the transgression would increase; but where sin increased, grace abounded all the more, so that, as sin reigned in death, even so grace would reign through righteousness to eternal life through Jesus Christ our Lord. (Rom 5:19-21)

The Son of Man will send forth His angels, and they will gather out of His kingdom all stumbling blocks, and those who commit lawlessness, and will throw them into the furnace of fire; in that place there will be weeping and gnashing of teeth. Then the righteous will shine forth as the sun in the kingdom of their Father. He who has ears, let him hear. (Matt 13:41-43)

Righteous behavior is a voluntary act as seen after human history has ended and appears represented as white robes.

Let us rejoice and be glad and give the glory to Him, for the marriage of the Lamb has come and His bride has made herself ready." It was given to her to clothe herself in fine linen, bright and clean; for the fine linen is the righteous acts of the saints. (Rev 19:7-8)

Blessed are those who wash their robes, so that they may have the right to the tree of life, and may enter by the gates into the city. (Rev 22:14)

While Jesus Christ encourages righteous behavior, the greatest joy in heaven is pursuing God's righteousness by conversion – faith in the One who God sent to atone for the sins of mankind.

So He told them this parable, saying, What man among you, if he has a hundred sheep and has lost one of them, does not leave the ninety-nine in the open pasture and go after the one which is lost until he finds it? When he has found it, he lays it on his shoulders, rejoicing. And when he comes home, he calls together his friends and his neighbors, saying to them, ‘Rejoice with me, for I have found my sheep which was lost!' I tell you that in the same way, there will be more joy in heaven over one sinner who repents than over ninety-nine righteous persons who need no repentance. (Luke 15:3-7)