What does it mean to be human? This is obviously a vast topic, but the best starting point in the Bible is Genesis 1, where God says, "Let us make man in our image, in our likeness, and let them rule over the fish of the sea and the birds of the air, over the livestock, over all the earth, and over all the creatures that move along the ground." After making man, he "blessed them and said to them, Be fruitful and multiply, fill the earth and subdue it, and rule over the fish of the sea and the birds of the air and over every living creature that moves on the ground."
From this passage, we see that what makes man human is that he is in God's image, as opposed to the animals, which are made after their own kinds. Clearly we are like God in some important way, unlike the animals, and clearly the the image of God is connected with ruling over the world. I think that it is best to say that the image of God is the likeness of man to God which enables him to rule. This consists of the intellectual ability to rule (planning, speech). It also consists of the character to rule, which is the main focus of the biblical discussion.
The image of God is also connected with sonship. Consider Genesis 4:1: "Adam had a son in his own likeness, in his own image." This is the only other place in the Bible where the phrase "in his own image and likeness" appears, and I think we are to understand the sons of God as those who bear the image of God. (Adam is explicitly called the "son of God" in Luke 3:37.) A son is to be like his father, and a son will do what his father does.
God created man as the keystone of his creation, and gave him the charge to rule, and in the same passage blessed all of creation. I think it is not a stretch to say that God gave man the charge to rule so that all creation would be blessed. But when Adam fell, he brought curse to the entire creation instead of blessing. Creation is now waiting in eager expectation for the unveiling of the sons of God (Rom 8:19-20) who will liberate it from its bondage to decay. What God is doing in redemption is creating a new humanity which will accomplish God's original plan to bring blessing. At key stages in redemptive history, the scripture is careful to connect the people of God with the blessing of creation.
When God calls Abram, he tells him to "leave your country, your people, and your father's household" (Genesis 12:1). He says it three times, so it must be important. There is more involved here than just leaving behind the idolatrous influence of his family (Joshuah 24:2). Leaving family behind was very unusual in the ancient world, where family gives a person his identity. Family ties define one's connection to the rest of the human race. (This is one reason why genealogies are so important in many cultures.) When Abram turns away from his family, he loses his connection to the rest of humanity, and that is the point: the family of Abraham will be a new humanity.
God promises to bless Abram and to make him a blessing, a clear allusion back to the blessing of creation. It is through Abraham that the blessing of God will reach the ends of the earth. Abram also will have many descendants, corresponding to God's blessing to Adam to "be fruitful and multiply". (In fact, his offspring "will be like the dust of the earth," which is perhaps an inverted allusion to God's curse on Adam that he would return to the dust.) Abraham's descendants will "rule over the gates of their enemies" (Genesis 22:17), corresponding to Adam's charter to rule. Abraham is given the land, which Paul understands to mean the whole earth (Romans 4:13), not just the land of Israel.
It is the descendants of Abraham, the children of Israel, who are referred to as the sons of God (Exodus 4:22). Note also the reiteration of blessing as part of the covenant (e.g., Deuteronomy 28) and the curses that follow if the people refuse their calling, just as with Adam. The people of Israel are intended to be the new humanity who accomplish God's purpose.
David is given a similar blessing: seed, a special relationship with God, authority, land. Note particularly 2 Sam 7:19, which can arguably be translated, "Is this your charter for humanity?" (See Kaiser, Toward an Old Testament Theology, pp. 152-155.) The kings of Israel are called "the son of God" (e.g., Psalm 2; 2 Samuel 7:14); they are the representative head of the new humanity, and the commission and the blessing are focused on them (e.g., Psalm 72: "All nations will be blessed through [the king]", referring back to Genesis 12).
Daniel 7 is God's plan for humanity. (It is a development of the same theme earlier in Daniel; chapter 2 is about humanity's destiny, and chapter 4 is about how proud humanity's boasts make it less than human.) Daniel 7 portrays the kingdoms of men as ravaging beasts--they are not even human. Men in their sinfulness and violence have sunk to the level of animals: they do not have the character to rule. But then "one like a son of man" is given the final authority to rule over the entire earth. The contrast between the man and the animals is strikingly reminiscent of Genesis 1. The angel interpreting the vision explains that the meaning of the son of man is that "the saints of the Most High will receive the kingdom and will possess it forever"--the truly human people will rule. This passage is particularly interesting because it highlights the divine likeness of the "son of man" (he comes with the clouds, and people worship him), and yet it connects it with the people of God. This tension between the "son of Man" as a divine person, and the "son of Man" as collectively the people of God is not resolved until the New Testament, where the "Son of Man" is in fact the Son of God, the exact image of God, and his people who are in his likeness and rule with him.
Though the Old Testament presents Israel as the new humanity, it is only as a type (a foreshadowing), not the reality. Israel, as is apparent from her record, behaves very much like the old humanity. The real change begins with Jesus and the coming of the Spirit, and is still continuing today. Of course, like Israel, the church does not often act as the true humanity, but with the Spirit it has the potential to do so, sometimes does now, and is guaranteed to do so in the future.
In the New Testament, it is Jesus who is the truly human person, because he is the exact image of God. A number of passages compare him with Adam (most obviously Romans 5 and 1 Corinthians 15). The gospels show how he was tempted as Adam was in a garden (and also in the wilderness, the opposite of a garden), but he obeyed. He wears a crown of thorns, symbolically bearing the curse. Pilate and the soldiers jeeringly proclaim him as king, but what they say is ironically true: Jesus is the ruler of the world. And Pilate, in another ironic act with much deeper significance than he realizes, brings Jesus out dressed as a king, and says, "Behold the man!" (John 19:5) That John intends this an allusion to Genesis 1 is likely given the other allusions to Genesis 1 nearby: this happens on the sixth day of the week, a day which is ended by Jesus' loud cry, "It is finished," and then comes the sabbath. The point is that in the death of Jesus, God has created the new humanity. Jesus has accomplished what God intended, and is ruling as God intended for man.
After his resurrection, Jesus breathes his Spirit into the disciples (John 20:22), which is an allusion to God breathing the breath of life into man in Genesis 2. (Also the Spirit came as a mighty rushing wind at Pentecost, which is likely an allusion to Genesis 1:2--now the whole world is being re-created, with the new humanity first rather than last.) After announcing that "all authority is given to me" as it was in part to Adam, he gives them the commission to "go into all the world and make disciples of all nations" (Matthew 28:19). This is essentially the same as saying "Be fruitful and multiply, fill the earth and subdue it."
Now we are the sons and daughters of God, and it has not yet appeared what we will be (1 John 3:1ff), but we will be like him when he appears. We, Jesus' brothers and sisters (Hebrews 2:11), are putting on the new man, which is being made in the image of our creator (Colossians 3:10). These and other New Testament passages about the image of God are primarily about having the character of God. We will one day rule the world with Jesus, the way God intended for Adam to rule; but we must have the character to rule.
The climax comes when Jesus has finally subdued the earth, as Adam was supposed to: "He must reign until he has put all his enemies under his feet.... For it says, 'he has put everything under his feet.'" (1 Corinthians 15:25ff, quoting Psalm 8.) Paul here takes what was said of humanity in general, that humans rule over the earth because God intended it that way, and applies it to Jesus in particular (as the new Adam), and explains it as the goal of history. The last Adam, unlike the first, "will himself be subject to him who put everything under him, so that God may be all in all." In the end, the curse that came from Adam's action is undone, the tree of life is again accessible, and the redeemed people of God rule over the earth as God intended (Rev. 22:1-5).
[Material for this was gathered primarily from W. Kaiser, Toward an Old Testament Theology, and N. T. Wright, The Resurrection of the Son of God, both of which are well worth reading for their insights into the development of key themes of the Bible from beginning to end.]