Praise be to Allah.

The one who took care of him was his grandfather ‘Abd al-Muttalib, then after his grandfather died, his paternal uncle Abu Taalib took care of him. Both of them died as disbelievers, but ‘Abd al-Muttalib died before the Prophet’s mission began. Abu Taalib died about ten years after the Prophet’s mission began, after refusing to enter Islam. 

Ibn Katheer said: The Messenger of Allaah (peace and blessings of Allaah be upon him) was with his mother Aaminah bint Wahb, then when she died, his grandfather ‘Abd al-Muttalib took care of him. He used to sit on his cushion and ‘Abd al-Muttaalib would not eat any food without saying “Bring me my son” and he would be brought to him. When ‘Abd al-Muttalib was on his deathbed, he told Abu Taalib to take care of the Messenger of Allaah (peace and blessings of Allaah be upon him). Ibn Ishaaq said: When the Messenger of Allaah (peace and blessings of Allaah be upon him) reached the age of eight years, his grandfather ‘Abd al-Muttalib ibn Haashim died. After his grandfather, the Messenger of Allaah (peace and blessings of Allaah be upon him) stayed with his paternal uncle Abu Taalib, in accordance with the last wishes of ‘Abd al-Muttalib, and because he was the full-brother of (the Prophet’s) father ‘Abd-Allaah – their mother was Faatimah bint ‘Amr ibn ‘Imraan ibn Makhzoom. So Abu Taalib was the one who looked after the affairs of the Messenger of Allaah (peace and blessings of Allaah be upon him). Abu Taalib had no wealth, but he loved him deeply, more than he loved his own children; he would not sleep unless he was his side, and when he went out he would go out with him, and he used to keep the best food for him.  

Al-Bidaayah wa’l-Nihaayah, vol. 2, p. 282.