Praise be to Allaah.
There is a difference between adoption and sponsoring orphans.
A – Adoption means that a man takes an orphan and makes him like one of his own children, calling him after him, so that the orphan is not allowed to marry one of the man’s daughters, and so the sons of the adoptive father are regarded as brothers of the orphan and his daughters are regarded as his sisters, and his (the father’s) sisters are regarded as his paternal aunts, and so on. This was one of the things that were done during the first Jaahiliyyah, and some of the Sahaabah carried the names of their adoptive fathers, as in the case of al-Miqdaad ibn al-Aswad whose real father’s name was ‘Amr, but he was called ibn (son of) al-Aswad, after the man who had adopted him.
This continued into the early days of Islam, until Allaah forbade that, according to a well-known story. Zayd ibn Haarithah was called Zayd ibn Muhammad, and he was the husband of Zaynab bint Jahsh, then Zayd divorced her.
It was narrated that Anas said: When Zaynab’s ‘iddah came to an end, the Messenger of Allaah (peace and blessings of Allaah be upon him) said to Zayd ibn Haarithah, “Go and tell her about me (that I want to marry her).” So he went to her and found her kneading dough. He said, “O Zaynab, good news. The Messenger of Allaah wants to marry you.” She said, “I will not do anything until I consult with my Lord.” Then she got up and went to her prayer-place, then the Messenger of Allaah (peace and blessings of Allaah be upon him) came and entered upon her.
Concerning this, Allaah revealed the words (interpretation of the meaning):
“And (remember) when you said to him (Zayd bin Haarithah the freed‑slave of the Prophet) on whom Allaah has bestowed grace (by guiding him to Islam) and you (O Muhammad too) have done favour (by manumitting him): ‘Keep your wife to yourself, and fear Allaah.’ But you did hide in yourself (i.e. what Allaah has already made known to you that He will give her to you in marriage) that which Allaah will make manifest, you did fear the people (i.e., their saying that Muhammad married the divorced wife of his manumitted slave) whereas Allaah had a better right that you should fear Him. So when Zayd had accomplished his desire from her (i.e. divorced her), We gave her to you in marriage, so that (in future) there may be no difficulty to the believers in respect of (the marriage of) the wives of their adopted sons when the latter have no desire to keep them (i.e. they have divorced them). And Allaah’s Command must be fulfilled”
[al-Ahzaab 33:37]
(Narrated by Muslim, 1428)
B – Allaah has forbidden adoption because it causes knowledge of people’s lineage to be lost, and we have been commanded to preserve people’s lineage.
It was narrated from Abu Dharr (may Allaah be pleased with him) that he heard the Prophet (peace and blessings of Allaah be upon him) say: “There is no man who knowingly calls himself after someone other than his father but he has committed kufr. Whoever claims to belong to people to whom he has no ties of blood, let him take his place in Hell.”
(Narrated by al-Bukhaari, 3317; Muslim, 61)
What is meant by “committed kufr” is that he has done an act of kufr, not that he is beyond the pale of Islam.
This may generate hatred and resentment between the adopted son and the children of the adoptive father, because it will cause them to lose out on something that is rightfully theirs, which will go to this orphan unlawfully who they know is not entitled to it as they are.
http://www.islam-qa.com/en/ref/5201/adoption