Adoption is the act of assuming parental care and responsibility of a person who is not one's own, and the converse, the act of appropriating-embracing the gift of parental care and heirdom given by another person who has taken us on as his/her own child. Adoption is the supreme gift Christians enjoy in Christ; it is the blessing that permits us, by the enabling of the Holy Spirit, to turn to God and call him, "Abba, Father." It is the blessing that enables us to be "heirs of God and co-heirs with Christ" (Rom 8:1). The importance of this gift is especially well captured by the following creed and Bible teachers:
Adoption is an act of God's free grace, whereby we are received into the number, and have a right to all the privileges of, the sons of God.
Westminster Shorter Catecism
Our understanding of Christianity cannot be better than our grasp of adoption. . . . If you want to judge how well a person understands Christianity, find out how much he makes of the thought of being God’s child, and having God as his Father. If this is not the thought that prompts and controls his worship and prayers and whole outlook on life, it means that he does not understand Christianity very well at all.
J. I. Packer
The image of "adoption" tells us that our relationship with God is based completely on a legal act by the Father. You don't "win" a father, and you don't "negotiate" for a parent. Adoption is a legal act on the part of the father—it is very expensive and costly only for him. There is nothing the son does to win or earn the status. It is simply received.
Timothy Keller
Adoption was a common practice a common practice in most human cultures, ancient and new. Ancient Hebrew society seems an exception, and the idea of adoption hardly appears in the Old Testament. Even in the one clear case in which a potential case of adoption is raised—Gen 15:3, in which Abram demurred to God, "You have given me no children; so a servant in my household will be my heir"—the custom was viewed with dissatisfaction. This, no doubt, has its effect on the theological language of the OT; God appears as Father rarely, and when it does, the language is largely metaphorical; Deut 1:31 (". . . in the desert. There you saw how the Lord your God carried you, as a father carries his son, all the way you went until you reached this place.") being a clear example. This is not to deny that the idea of adoption is absent in the OT. Paul, in discussing about the fate of the people of Israel in God's plan clearly understood that "theirs is the adoption as sons" (Rom 9:4). This idea, however, is present in the OT only, to use a physical chemical analogy, in "solution" rather than in "crystals."
In sharp contrast to the world of the Hebrews, adoption was common in the Greco-Roman world. Whereas most cultures, in especially in modern ones, adopt the child in infancy, it was the Roman custom for a man to adopt another adult male to carry on the family name and to serve as his heir. The apostles and their audience would have been well-familiar with the two famous adoptions that formed the background to the New Testament. There was, first, the adoption of the 19-year old Gaius Octavian (b.63 BC, d.14 AD) by Julius Caesar. Octavian — better known by his later name Caesar Augustus — was the Roman Emperor who confirmed Herod the Great as king over Judea, and during whose reign Jesus was born. Octavian was succeeded, in turn, by his adopted son, Tiberius (b.42 BC, d.37 AD) (It was in honour of these two emperors that the Herods founded, and named after them, the cities of Caesarea — where Paul was imprisoned for a short while before being sent on to Rome (Acts 23:2326:32) — and Tiberias, the city on shore of the Sea of Galilee.) Against this background of the Roman emperors adopting another to reign in his place after he is gone, it is little surprising that the New Testament speaks of the blessing of adoption, together with its responsibility and benefits, with no preamble.
The doctrine of adoption in the New Testament is conveyed via a number of different words. Paul uses the noun υἱοθεσία, huiothesia, literally "sonship' (from huios, 'son'), both in Rom 8:23 and Eph 1:4-6. The Greek word was a legal term that declares that the adoptee now enjoys the full rights, privileges, and power of being an heir. The most powerful and personally evocative word for adoption in the NT comes, however from Jesus; that word is, in their order of appearance, "your Father," then "our Father, who is in heaven."
We would not have known God as our Father if not for Jesus's teaching about it.