Summary: The word "adoption" is an action noun, so it acts like a verb, i.e. the Holy Spirit performs the act of adopting. In Roman law an adopted child had the exact same position and privileges as a natural born child, e.g. legal standing, or inheritance.
Purpose: To show what spiritual adoption is and what a great privilege it is.
INTRODUCTION: ADOPTION huiothesia ^5206^, from huios, "a son," and thesis, "a placing," akin to tithemi, "to place," signifies the place and condition of a son given to one to whom it does not naturally belong. The word is used by the apostle Paul only. 
The word "adoption" is an action noun, so it acts like a verb, i.e. the Holy Spirit performs the act of adopting.
In Roman law an adopted child had the exact same position and privileges as a natural born child, e.g. legal standing, or inheritance.
I. Why we need God to Adopt us
We aren't God's children by nature. Eph 2:1 2 1 As for you, you were dead in your transgressions and sins, 2 in which you used to live when you followed the ways of this world and of the ruler of the kingdom of the air, the spirit who is now at work in those who are disobedient. (NIV) Eph 2:4 5 4 But because of his great love for us, God, who is rich in mercy, 5 made us alive with Christ even when we were dead in transgressions it is by grace you have been saved. (NIV)
Gal 4:4 5 4 But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law, 5 To redeem them that were under the law, that we might receive the adoption of sons. (KJV)
Notice verse five in the NIV: Gal 4:5 to redeem those under law, that we might receive the full rights of sons. (NIV)
By the way, Jesus is the Son of God; He is never referred to as adopted.
II. The Privilege of Adoption
A. It gives us a new Standing with God
Gal 4:6 Because you are sons, God sent the Spirit of his Son into our hearts, the Spirit who calls out, "Abba, Father." (NIV)
B. It gives us a new Nature
2Cor 5:17 Therefore, if anyone is in Christ, he is a new creation; the old has gone, the new has come! (NIV)
2Pet 1:3 4 3 His divine power has given us everything we need for life and godliness through our knowledge of him who called us by his own glory and goodness. 4 Through these he has given us his very great and precious promises, so that through them you may participate in the divine nature and escape the corruption in the world caused by evil desires. (NIV)
III. The Process of Adoption
A. It is given by the Holy Spirit
Notice the word "received" in Rom 8:15 For you did not receive a spirit that makes you a slave again to fear, but you received the Spirit of sonship. And by him we cry, "Abba, Father." (NIV)
John 1:12 Yet to all who received him, to those who believed in his name, he gave the right to become children of God (NIV)
"Right" refers to legal right. Imagine having such a great right!
B. It was planned by God through Jesus Christ
Eph 1:5 he predestined us to be adopted as his sons through Jesus Christ, in accordance with his pleasure and will (NIV)
John 1:13 children born not of natural descent, nor of human decision or a husband's will, but born of God. (NIV)
I.e. our birth is not our decision or someone else's, only God's will.
IV. The Results of Adoption
A. We gain the same fellowship that Christ has with the Father
Mark 14:36 "Abba, Father," he said, "everything is possible for you. Take this cup from me. Yet not what I will, but what you will." (NIV)
Rom 8:32 He who did not spare his own Son, but gave him up for us all how will he not also, along with him, graciously give us all things? (NIV)
B. We gain the same inheritance that Christ has
Gal 4:6 7 6 Because you are sons, God sent the Spirit of his Son into our hearts, the Spirit who calls out, "Abba, Father." 7 So you are no longer a slave, but a son; and since you are a son, God has made you also an heir. (NIV)
Rom 8:16 17 16 The Spirit himself testifies with our spirit that we are God's children. 17 Now if we are children, then we are heirs heirs of God and co heirs with Christ, if indeed we share in his sufferings in order that we may also share in his glory. (NIV)
C. We gain a Longing to be with Christ
Rom 8:22 23 22 We know that the whole creation has been groaning as in the pains of childbirth right up to the present time. 23 Not only so, but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for our adoption as sons, the redemption of our bodies. (NIV)