Adoption is the giving to any one the name and place and privileges of a son who is not a son by birth.  
(1.) Natural. Thus Pharaoh's daughter adopted Moses (Ex 2:10), and Mordecai Esther (Es 2:7).
(2.) National. God adopted Israel (Ex 4:22; De 7:6; Ho 11:1; Ro 9:4).
(3.) Spiritual. An act of God's grace by which he brings men into the number of his redeemed family, and makes them partakers of all the blessings he has provided for them. Adoption represents the new relations into which the believer is introduced by justification, and the privileges connected therewith, viz., an interest in God's peculiar love (Joh 17:23; Ro 5:5-8), a spiritual nature (2Pe 1:4; Joh 1:13), the possession of a spirit becoming children of God (1Pe 1:14; 2Jo 1:13; Ro 8:15-21; Ga 5:1; Heb 2:15), present protection, consolation, supplies (Lu 12:27-32; Joh 14:18; 1Co 3:21-23; 2Co 1:4), fatherly chastisements (Heb 12:5-11), and a future glorious inheritance (Ro 8:17,23; Jas 2:5; Php 3:21).

Source: Easton's Bible Dictionary