Bismillah.
A simple and direct answer to who Abu Talib was (according to answers.com):
(d. ca. 619) Muhammad's uncle, protector, and guardian after the death of his parents. According to tradition, he was present when Muhammad was recognized as a prophet by the Christian monk Bahira. Did not convert to Islam but offered his own protection, as well as that of the Banu Hashim clan, to Muhammad when his preaching was opposed, permitting survival of the early Muslim community.
Narrated Said bin Al-Musaiyab from his father:
When the time of the death of Abu Talib approached, Allah's Apostle went to him and found Abu Jahl bin Hisham and 'Abdullah bin Abi Umaiya bin Al-Mughira by his side. Allah's Apostle said to Abu Talib, "O uncle! Say: None has the right to be worshipped but Allah, a sentence with which I shall be a witness (i.e. argue) for you before Allah. Abu Jahl and 'Abdullah bin Abi Umaiya said, "O Abu Talib! Are you going to denounce the religion of Abdul Muttalib?" Allah's Apostle kept on inviting Abu Talib to say it (i.e. 'None has the right to be worshipped but Allah') while they (Abu Jahl and Abdullah) kept on repeating their statement till Abu Talib said as his last statement that he was on the religion of Abdul Muttalib and refused to say, 'None has the right to be worshipped but Allah.' Then Allah's Apostle said, "I will keep on asking Allah's forgiveness for you unless I am forbidden (by Allah) to do so." So Allah revealed (the verse) concerning him (i.e. It is not fitting for the Prophet and those who believe that they should invoke (Allah) for forgiveness for pagans even though they be of kin, after it has become clear to them that they are companions of the fire (9.113).
﴿إِنَّكَ لاَ تَهْدِى مَنْ أَحْبَبْتَ وَلَـكِنَّ اللَّهَ يَهْدِى مَن يَشَآءُ وَهُوَ أَعْلَمُ بِالْمُهْتَدِينَ﴾
"Verily, you guide not whom you like, but Allah guides whom He wills. And He knows best those who are the guided." [Al-Qasas: 56]
Ibn Kathir says in his tafsir:
Allah says to His Messenger : `O Muhammad:
﴿لاَ تَهْدِى مَنْ أَحْبَبْتَ﴾
(Verily, you guide not whom you like)' meaning, `the matter does not rest with you; all that you have to do is convey the Message, and Allah will guide whom He wills, and His is the ultimate wisdom,' as He says:
﴿لَّيْسَ عَلَيْكَ هُدَاهُمْ وَلَـكِنَّ اللَّهَ يَهْدِى مَن يَشَآءُ﴾
(Not upon you is their guidance, but Allah guides whom He wills.) (2:272)
﴿وَمَآ أَكْثَرُ النَّاسِ وَلَوْ حَرَصْتَ بِمُؤْمِنِينَ ﴾
(And most of mankind will not believe even if you desire it eagerly.) (12:103) This Ayah is even more specific than the following:
﴿إِنَّكَ لاَ تَهْدِى مَنْ أَحْبَبْتَ وَلَـكِنَّ اللَّهَ يَهْدِى مَن يَشَآءُ وَهُوَ أَعْلَمُ بِالْمُهْتَدِينَ ﴾
(Verily, you guide not whom you like, but Allah guides whom He wills. And He knows best those who are the guided.) meaning: Allah knows best who deserves to be guided and who deserves to be misguided. It was recorded in the Two Sahihs that this Ayah was revealed concerning Abu Talib, the paternal uncle of the Messenger of Allah . He used to protect the Prophet, support him and stand by him. He loved the Prophet dearly, but this love was a natural love, i.e., born of kinship, not a love that was born of the fact that he was the Messenger of Allah . When he was on his deathbed, the Messenger of Allah called him to Faith and to enter Islam, but the decree overtook him and he remained a follower of disbelief, and Allah's is the complete wisdom. Az-Zuhri said: "Sa`id bin Al-Musayyib narrated to me that his father, Al-Musayyib bin Hazan Al-Makhzumi, may Allah be pleased with him, said: "When Abu Talib was dying, the Messenger of Allah came to him and found Abu Jahl bin Hisham and `Abdullah bin Abi Umayyah bin Al-Mughirah with him. The Messenger of Allah said:
«يَا عَمِّ قُلْ: لَا إِلَهَ إِلَّا اللهُ، كَلِمَةً أُحَاجُّ لَكَ بِهَا عِنْدَ الله»
(O my uncle, say La ilaha illallah, a word which I may use as evidence in your favor before Allah ﴿in the Hereafter﴾.) Abu Jahl bin Hisham and `Abdullah bin Abi Umayyah said: `O Abu Talib, will you leave the religion of `Abdul-Muttalib' The Messenger of Allah kept urging him to say La ilaha illallah, and they kept saying, `Will you leave the religion of `Abdul-Muttalib' -- until, at the very end, he said that he was on the religion of `Abdul-Muttalib, and he refused to say La ilaha illallah. The Messenger of Allah said:
«وَاللهِ لَأَسْتَغْفِرَنَّ لَكَ مَا لَمْ أُنْهَ عَنْك»
(By Allah, I shall certainly seek forgiveness for you unless I am told not to.) Then Allah revealed:
﴿مَا كَانَ لِلنَّبِىِّ وَالَّذِينَ ءَامَنُواْ أَن يَسْتَغْفِرُواْ لِلْمُشْرِكِينَ وَلَوْ كَانُواْ أُوْلِى قُرْبَى﴾
(It is not (proper) for the Prophet and those who believe to ask Allah's forgiveness for the idolators, even though they be of kin) (9:113). And there was revealed concerning Abu Talib the Ayah,
﴿إِنَّكَ لاَ تَهْدِى مَنْ أَحْبَبْتَ وَلَـكِنَّ اللَّهَ يَهْدِى مَن يَشَآءُ﴾
(Verily, you guide not whom you like, but Allah guides whom He wills.)'' This was recorded (by Al-Bukhari and Muslim) from the Hadith of Az-Zuhri.
It is not allowed to pray for the forgiveness of non-Muslims after their death. No funeral prayer for them either. And no burying them according to the Islamic way.
Wassalam.
