But to all who received him, who believed in his name, he gave ἐξουσία to become children of God.The Greek word ἐξουσία (exousia) can be interpreted in at least two ways, either as authority or power. Of course, though these concepts are very closely related, in many ways they come apart. Authority is something extrinsic and relational: a person's authority is just the way she relates to other people, the rights she has and the expectations she can have of others. On the other hand, power is something intrinsic to a person, something which is in an important sense "within her."
Consider the following example to see the difference between the two. Suppose I lift very heavy weights for a long time, and thus have it within it my power to move parked vehicles out of their spots. It doesn't follow from my having this power that therefore I have the authority to make room for my own car in this manner! Or suppose, on the other hand, that I am a professor at the university, and have temporarily lost my voice. This means that I have the authority to lecture my students on some topic or other, but I don't have the power to do so. In these ways, then, it is clear that power and authority come apart.
Now the question arises: how should we translate ἐξουσία as it appears in this verse of the Gospel? The translation we choose will undoubtedly say much about how we understand the nature of salvation through Christ. Does Christ give us authority to become children of God? If this is the way we choose, then we will largely conceive of salvation in external, legal terms: salvation will have more to do with the relations we stand in relative to God, rather than what's happening on the inside of us, in our being. On the other hand, if we choose to translate ἐξουσία as power, then salvation will largely be understood in terms of a transformation of our being in a certain direction. Becoming "children of God", on this view of things, is not merely a change of relation to God, but rather becoming different sorts of persons.
Of course, these two understandings are not mutually exclusive. All Reformed Protestants, for example, who in my judgment will tend towards understanding ἐξουσία as authority, will accept that our relational change vis-a-vis God will also be followed by a change within us, by virtue of which we will be conformed to the image of his Son. Likewise, all Catholics and Orthodox, who might be likelier to understand ἐξουσία as power, will agree that there are important relational changes involved in the process of salvation, as well. Yet they will differ on the understanding of what it means, primarily, to be a "child of God."
The Reformed, who understand ἐξουσία as authority, will emphasize that being a child of God is primarily a relational stance. Human sinners, even granting all their sins which may be grave, are nevertheless children of God because they stand in a certain relation to God through Christ. They are not less children of God because of their sins, because this relational stance is mediated through Christ's perfect righteousness which is imputed to them. But for the Catholics and Orthodox, understanding ἐξουσία as power, being a child of God is primarily about being a certain way: loving God and obeying his commandments. This relation can be broken if a person commits a mortal sin, after which time the relation of this person to God must be repaired through the sacrament of reconciliation. Salvation will be first and foremost a matter of receiving the power to be a child of God through loving God and living in obedience to his commandments.
So there are very deep and important theological issues involved in the interpretation of this one verse. The NRSV interprets ἐξουσία as power, whereas the ESV, NIV, and NASB interpet ἐξουσία as right. Which is the right way to go?