These two Ayas are the key source:

وَلَا تَنكِحُوا مَا نَكَحَ آبَاؤُكُم مِّنَ النِّسَاءِ إِلَّا مَا قَدْ سَلَفَ ۚ إِنَّهُ كَانَ فَاحِشَةً وَمَقْتًا وَسَاءَ سَبِيلًا

And do not marry those [women] whom your fathers married, except what has already occurred. Indeed, it was an immorality and hateful [to Allah ] and was evil as a way.

حُرِّمَتْ عَلَيْكُمْ أُمَّهَاتُكُمْ وَبَنَاتُكُمْ وَأَخَوَاتُكُمْ وَعَمَّاتُكُمْ وَخَالَاتُكُمْ وَبَنَاتُ الْأَخِ وَبَنَاتُ الْأُخْتِ وَأُمَّهَاتُكُمُ اللَّاتِي أَرْضَعْنَكُمْ وَأَخَوَاتُكُم مِّنَ الرَّضَاعَةِ وَأُمَّهَاتُ نِسَائِكُمْ وَرَبَائِبُكُمُ اللَّاتِي فِي حُجُورِكُم مِّن نِّسَائِكُمُ اللَّاتِي دَخَلْتُم بِهِنَّ فَإِن لَّمْ تَكُونُوا دَخَلْتُم بِهِنَّ فَلَا جُنَاحَ عَلَيْكُمْ وَحَلَائِلُ أَبْنَائِكُمُ الَّذِينَ مِنْ أَصْلَابِكُمْ وَأَن تَجْمَعُوا بَيْنَ الْأُخْتَيْنِ إِلَّا مَا قَدْ سَلَفَ ۗ إِنَّ اللَّهَ كَانَ غَفُورًا رَّحِيمًا

Prohibited to you [for marriage] are your mothers, your daughters, your sisters, your father’s sisters, your mother’s sisters, your brother’s daughters, your sister’s daughters, your [milk] mothers who nursed you, your sisters through nursing, your wives’ mothers, and your step-daughters under your guardianship [born] of your wives unto whom you have gone in. But if you have not gone in unto them, there is no sin upon you. And [also prohibited are] the wives of your sons who are from your [own] loins, and that you take [in marriage] two sisters simultaneously, except for what has already occurred. Indeed, Allah is ever Forgiving and Merciful.
(Surat An-Nisā’ Aya 22-23.)

Allah prohibited 14 classes of women, 7 due to blood relationship, and 7 for others.

Those because of blood relationship are:

  • Mothers: every female ancestor for you no matter how high her level is (note that she might on the side of your father too, for example, your father’s mother).
  • Daughters: every female descendant of you no matter how low her level is (again, note that she might be on the side of your son, for example, your son’s daughter).
  • Sisters: every female that share you your father or mather (or both of them).
  • Aunts: both
    • Sisters of every male ancestor for you (even on the side of your mother, such as the sister of your mother’s father), and
    • Sisters of every female ancestor for you (even on the side of your father, such as the sister of your father’s mother).
  • Nieces: every female descendant of your brothers and sisters, no matter how low her level is (this count 2 out of 7). This completes the list of women prohibited because of blood relationship.

Now we come to those prohibited because of milk relationship (breastfeeding), there is a basic hadith here:

The Prophet said about Hamza’s daughter, “I am not legally permitted to marry her, as foster relations are treated like blood relations (in marital affairs). She is the daughter of my foster brother.”

حَدَّثَنَا مُسْلِمُ بْنُ إِبْرَاهِيمَ، حَدَّثَنَا هَمَّامٌ، حَدَّثَنَا قَتَادَةُ، عَنْ جَابِرِ بْنِ زَيْدٍ، عَنِ ابْنِ عَبَّاسٍ ـ رضى الله عنهما ـ قَالَ قَالَ النَّبِيُّ صلى الله عليه وسلم فِي بِنْتِ حَمْزَةَ ‏ “‏ لاَ تَحِلُّ لِي، يَحْرُمُ مِنَ الرَّضَاعِ مَا يَحْرُمُ مِنَ النَّسَبِ، هِيَ بِنْتُ أَخِي مِنَ الرَّضَاعَةِ ‏”‏‏.‏

Sahih al-Bukhari 2645 (with many, many other riwayat [versions] ).

And, depending on that hadith, there is a basic rule in Shariaa:

يحرم من الرضاع ما يحرم من النسب

What’s prohibited (for marriage) because of milk relationship are the same as what’s prohibited because of blood relationship.

Other translation: Marriage prohibition because of milk relationship is similar to marriage prohibition because of blood relationship.

Now if a woman breastfeeds a boy, then she is considered (in marital affairs) as his own mother, and blood-relationship prohibition rules are applied with all of her family. And the husband of this woman at the time she breastfed the boy is considered (in marital affairs) as his own father, and (again) blood-relationship prohibition rules are applied with all of his family.

Also milk sisters are prohibited, they’re every female breastfed by your (blood) mother or your mother in law (your father’s wife). And she is considered (in maritial affairs) as your own sister.

But, the amount of breastfeeding that makes these rules applied is not the same for all scholars, and I’m not going to explain this now. (general speaking two pleasing breadfeeds are enough to make these rules applied).

Now we come to those prohibited because of affinity, they’re:

  • Wife’s mothers: every female ancestor of the wife. And they become prohibited just if the marriage contract is made (even before any intercourse).
  • Wife’s daughters: every female descendant of man’s wife from another husband. At the condition of making an intercourse with this wife, i.e, marriage contract is not enough to prohibit these daughters, an intercourse is needed to make the prohibition applied.
  • Daughters in law: the wife of any male descendant, no matter how low her level is.
  • Milk Sons’ wives: wives of man’s milk sons (the rule of milk prohibition applies here).
  • Mother in law: the wife of any male ancestor, no matter how high his level is. Also these three classes have some differences in scholars’ views, in affairs of intercourse or just marriage contract… etc. But again, I’m not going to discuss these now.
  • Sisters of current wives: a man is not allowed to marry two sisters at the same time, if he marry a woman, then all of her sisters are temporarily prohibited for him. If she die, or they divorce, then her sisters become un-prohibited for him, he can marry any of them (after the divorce period عدة of the wife is over).

And finally, of course he can’t marry a married woman! if she divorce, then he can marry her after her divorce period عدة is over.