A woman cannot be married without a guardian (wali); she cannot give herself away in marriage.
Any marriage without a wali is legally invalid [1]
[1.'Legally' here means in Islamic Law-Translator]
The woman's guardians (awliya') are those who are related to her from her father's side.
Jurists stipulate seven requirements for a valid guardian. If they are not met, a person does not qualify to be the guardian
Seven requirements for a valid guardian
1. He must be legally responsible: this excludes the insane and the minor
2. He must be male: a woman cannot give away another woman in marriage
3. He must be competent : he must know who and what is suitable for the marriage, as per the requirements of the situation
4. He must be of the same religion: an unbeliever cannot have guardianship of a Muslim woman
5. He must be upright and trustworthy [2] [2.i.e. Not a fasiq (sinner)ı Translator]
6. He must be a free man: a slave does not have wilayah over himself let alone anyone else
7. He must come in order: the more distant relative cannot marry off a woman if someone closer is available, unless there is a legitimate reason (see next point)
The guardianship (wilayah) is in order of closeness in relation, so the more distant relative (further down the list) cannot be taken as a walı unless the closer relative is not available: either because he is not present or because he prevents her from marrying (for no legitimate reason).
If more than one potential guardian is available and they are in equal place,the guardianship goes to the one who is approved by the woman and thus... ... if this chosen guardian marries her off, the marriage is valid even if he is younger than his equivalent in the list order.
The order of who is to be the woman's guardian
1. Her father
2. His agent if he has appointed one
3. Her paternal grandfather
4. Her son
5. His sons, and all his living male progeny [3] (grandsons, great-grandsons etc). [3. There is a difference of opinion on sons being awliyı'; most scholars say sons have the right to guardianship. This is discussed later]
6. Her full brothers
7. Her paternal half brothers
8. The sons of her full brothers
9. The sons of her paternal half brothers
10. Her full paternal uncles (i.e. brothers of her father who share both parents with him)
11. Her paternal half uncles (i.e.her father's half brothers from his father)
12. The sons of her paternal full uncles
13. The sons of her paternal half uncles
14. Those who are closest in paternal relation as per the rules of inheritance
15. Then the mawlı who is a benefactor e.g. in the case of a freed slave woman,the owner who set her free