What is adoption?
Adoption is an act of God's free grace1, whereby we are received into the number, and have a right to all the privileges, of the sons of God2.
Scripture proofs
The Westminster Assembly's proof texts (KJV). The bracketed numbers in the answer above mark the clause each set of references supports; expand a row to read the verses.
1 1 John 3:1
Behold, what manner of love the Father hath bestowed upon us, that we should be called the sons of God: therefore the world knoweth us not, because it knew him not.
2 John 1:12; Rom. 8:17
But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:
And if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together.
This doctrine across the Standards
Where the Confession and the Westminster Larger Catechism treat the same matter, so the question can be studied alongside its parallel statements.
In the Westminster Confession
XII.1 All those that are justified God vouchsafeth, in and for his only Son Jesus Christ, to make partakers …
All those that are justified God vouchsafeth, in and for his only Son Jesus Christ, to make partakers of the grace of adoption; by which they are taken into the number, and enjoy the liberties and privileges of the children of God; have his name put upon them; receive the Spirit of adoption; have access to the throne of grace with boldness; are enabled to cry, Abba, Father; are pitied, protected, provided for, and chastened by him as by a father; yet never cast off, but sealed to the day of redemption, and inherit the promises, as heirs of everlasting salvation.
→ XII.1 · Of AdoptionIn the Westminster Larger Catechism
Q.74 What is adoption?
Adoption is an act of the free grace of God, in and for his only Son Jesus Christ, whereby all those that are justified are received into the number of his children, have his name put upon them, the Spirit of his Son given to them, are under his fatherly care and dispensations, admitted to all the liberties and privileges of the sons of God, made heirs of all the promises, and fellow heirs with Christ in glory.
→ Open Q.74