:wasalamex
A person doesn't adopt a person in islam, in the sense that they don't get a child from another parent and say that its really their child. I.e. if you found a child called Abdullah, you couldn't adopt him and say he is now Abdullah ibn [son of] Hamayun. Instead, the child has to have the surname of his real biological father.
During the life of Prophet Muhammad, this was the case amongst the arabs. They adopted someone and called it theirs. i.e. Prophet Muhammad had a adopted son called Zayd, who was called Zayd ibn Muhammad by the people, until Allah forbade it [in the beginning verses of surah ahzab (33)] - so he was called Zayd ibn Harithah [the real father of Zayd.]
Fostering is allowed where you lookafter the child and provide for them, but you don't give them your name. Instead the child keeps their parents name and you bring them up. However, when they become mature [i.e. hit puberty] - they may be a non mahram, i.e. if you fostered a girl, when she hit puberty - she would be a stranger to you, you could potentially marry her [since she's not your real biological daughter] and therefore you can not see her except with full hijab etc. like any non mahram woman. The same with a fostered son being a non mahram for your wife - when he hits puberty.
This can be avoided though if the child is below 2 years when you foster them, in which the wife may give the child 5 full breastfeedings (so the child has 5 full milk meals and is full each meal). Then the child becomes your 'son/daughter' through the milk of breastfeeding, so when they grow older - they do not need to wear hijab infront of you. They are like a brother and sister to your biological children.
Also read;
An Adopted Child should be named after His Real Father
http://tafsir.com/default.asp?sid=33&tid=41259
Breastfeeding to make child a mahram
http://www.islam-qa.com/en/ref/97087/breastfeed%20five
Allah knows best.