There are some verses in Quran about relationship with unbelievers:
لَّا يَتَّخِذِ الْمُؤْمِنُونَ الْكَافِرِينَ أَوْلِيَاءَ مِن دُونِ الْمُؤْمِنِينَ ۖ وَمَن يَفْعَلْ ذَٰلِكَ فَلَيْسَ مِنَ اللَّهِ فِي شَيْءٍ إِلَّا أَن تَتَّقُوا مِنْهُمْ تُقَاةً ۗ وَيُحَذِّرُكُمُ اللَّهُ نَفْسَهُ ۗ وَإِلَى اللَّهِ الْمَصِيرُ ٣:٢٨
Let not the believers Take for friends or helpers Unbelievers rather than believers: if any do that, in nothing will there be help from Allah: except by way of precaution, that ye may Guard yourselves from them. But Allah cautions you (To remember) Himself; for the final goal is to Allah.
يَا أَيُّهَا الَّذِينَ آمَنُوا لَا تَتَّخِذُوا الْكَافِرِينَ أَوْلِيَاءَ مِن دُونِ الْمُؤْمِنِينَ ۚ أَتُرِيدُونَ أَن تَجْعَلُوا لِلَّهِ عَلَيْكُمْ سُلْطَانًا مُّبِينًا ٤:١٤٤
O ye who believe! Take not for friends unbelievers rather than believers: Do ye wish to offer Allah an open proof against yourselves?
الَّذِينَ يَتَّخِذُونَ الْكَافِرِينَ أَوْلِيَاءَ مِن دُونِ الْمُؤْمِنِينَ ۚ أَيَبْتَغُونَ عِندَهُمُ الْعِزَّةَ فَإِنَّ الْعِزَّةَ لِلَّهِ جَمِيعًا ٤:١٣٩
Yea, to those who take for friends unbelievers rather than believers: is it honour they seek among them? Nay,- all honour is with Allah.
يَا أَيُّهَا الَّذِينَ آمَنُوا لَا تَتَّخِذُوا عَدُوِّي وَعَدُوَّكُمْ أَوْلِيَاءَ تُلْقُونَ إِلَيْهِم بِالْمَوَدَّةِ وَقَدْ كَفَرُوا بِمَا جَاءَكُم مِّنَ الْحَقِّ يُخْرِجُونَ الرَّسُولَ وَإِيَّاكُمْ ۙ أَن تُؤْمِنُوا بِاللَّهِ رَبِّكُمْ إِن كُنتُمْ خَرَجْتُمْ جِهَادًا فِي سَبِيلِي وَابْتِغَاءَ مَرْضَاتِي ۚ تُسِرُّونَ إِلَيْهِم بِالْمَوَدَّةِ وَأَنَا أَعْلَمُ بِمَا أَخْفَيْتُمْ وَمَا أَعْلَنتُمْ ۚ وَمَن يَفْعَلْهُ مِنكُمْ فَقَدْ ضَلَّ سَوَاءَ السَّبِيلِ ٦٠:١
O ye who believe! Take not my enemies and yours as friends (or protectors),- offering them (your) love, even though they have rejected the Truth that has come to you, and have (on the contrary) driven out the Prophet and yourselves (from your homes), (simply) because ye believe in Allah your Lord! If ye have come out to strive in My Way and to seek My Good Pleasure, (take them not as friends), holding secret converse of love (and friendship) with them: for I know full well all that ye conceal and all that ye reveal. And any of you that does this has strayed from the Straight Path.
إِن يَثْقَفُوكُمْ يَكُونُوا لَكُمْ أَعْدَاءً وَيَبْسُطُوا إِلَيْكُمْ أَيْدِيَهُمْ وَأَلْسِنَتَهُم بِالسُّوءِ وَوَدُّوا لَوْ تَكْفُرُونَ ٦٠:٢
If they were to get the better of you, they would behave to you as enemies, and stretch forth their hands and their tongues against you for evil: and they desire that ye should reject the Truth.
يَا أَيُّهَا الَّذِينَ آمَنُوا لَا تَتَوَلَّوْا قَوْمًا غَضِبَ اللَّهُ عَلَيْهِمْ قَدْ يَئِسُوا مِنَ الْآخِرَةِ كَمَا يَئِسَ الْكُفَّارُ مِنْ أَصْحَابِ الْقُبُورِ ٦٠:١٣
O ye who believe! Turn not (for friendship) to people on whom is the Wrath of Allah, of the Hereafter they are already in despair, just as the Unbelievers are in despair about those (buried) in graves
Who are these unbelievers? What properties and characteristics they must have? Does any unbeliever is contained in these verses?
I know that there are some verses which maybe explain it but I'm not sure these verses are about them:
لَّا يَنْهَاكُمُ اللَّهُ عَنِ الَّذِينَ لَمْ يُقَاتِلُوكُمْ فِي الدِّينِ وَلَمْ يُخْرِجُوكُم مِّن دِيَارِكُمْ أَن تَبَرُّوهُمْ وَتُقْسِطُوا إِلَيْهِمْ ۚ إِنَّ اللَّهَ يُحِبُّ الْمُقْسِطِينَ٦٠:٨
Allah forbids you not, with regard to those who fight you not for (your) Faith nor drive you out of your homes, from dealing kindly and justly with them: for Allah loveth those who are just.
إِنَّمَا يَنْهَاكُمُ اللَّهُ عَنِ الَّذِينَ قَاتَلُوكُمْ فِي الدِّينِ وَأَخْرَجُوكُم مِّن دِيَارِكُمْ وَظَاهَرُوا عَلَىٰ إِخْرَاجِكُمْ أَن تَوَلَّوْهُمْ ۚ وَمَن يَتَوَلَّهُمْ فَأُولَٰئِكَ هُمُ الظَّالِمُونَ ٦٠:٩
Allah only forbids you, with regard to those who fight you for (your) Faith, and drive you out of your homes, and support (others) in driving you out, from turning to them (for friendship and protection). It is such as turn to them (in these circumstances), that do wrong.