Allah says that among His signs is that He created for you spouses to find tranquillity in them, and placed between you affection and mercy (30:21). A valid marriage needs the consent of both parties, the guardian (wali) of the woman, witnesses, and the mahr. The Prophet ﷺ said that a previously married woman is not married until she is consulted and a virgin until her permission is asked.
1
The requirements
The consent of the bride and the groom. Forcing a woman to marry is not valid.
The guardian of the woman: the Prophet ﷺ said that there is no marriage without a guardian (reported by Abu Dawud and graded sahih by many).
Two just witnesses and the announcement of the marriage.
The mahr, which belongs to the wife. The Prophet ﷺ taught a man to give something, even an iron ring, or teach some of the Quran he knew.
2
The prohibited marriages
A woman who is forbidden by relation (the mothers, daughters, sisters and so on), by breastfeeding or by marriage (4:23).
Two sisters at the same time, and a woman at the same time as her aunt.
A Muslim woman marrying a non-Muslim man. A Muslim man may marry a chaste woman of the People of the Book (5:5).
A woman in her iddah (waiting period).
Where the schools differ
The four Sunni schools agree on the main ruling above and differ in these points. No school is preferred here.
The guardian. the majority (Maliki, Shafi'i, Hanbali) say there is no marriage without a guardian. The Hanafi school allows an adult, sane woman to contract her own marriage, though it is recommended to involve the guardian.
Witnesses. the Hanafi, Shafi'i and Hanbali schools require two witnesses at the contract, and the Maliki school accepts the public announcement if witnesses were not present at the contract itself.
And give unto the women (whom ye marry) free gift of their marriage portions; but if they of their own accord remit unto you a part thereof, then ye are welcome to absorb it (in your wealth).
And of His signs is this: He created for you helpmeets from yourselves that ye might find rest in them, and He ordained between you love and mercy. Lo! herein indeed are portents for folk who reflect.
And marry such of you as are solitary and the pious of your slaves and maid-servants. If they be poor, Allah will enrich them of His bounty. Allah is of ample means, Aware.
From the Sunnah
1Sahih al-Bukhari 5136Authentic collection
The point of the hadith: A previously married woman is not given in marriage until she is consulted, and a virgin until her permission is asked.
The point of the hadith: The Prophet ﷺ told a man who wished to marry to look for something to give, even an iron ring, and when he found nothing, married him for the Quran he knew.
Marriage in Islam: the conditions of a valid contract
Allah says that among His signs is that He created for you spouses to find tranquillity in them, and placed between you affection and mercy (30:21). A valid marriage needs the consent of both parties, the guardian (wali) of the woman, witnesses, and the mahr. The Prophet ﷺ said that a previously married woman is not married until she is consulted and a virgin until her permission is asked.
Is this a fatwa?
No. It is a summary of well-known rulings with their evidence. For your own case, ask a qualified scholar.
What if my school differs from this answer?
Follow the view of the school you follow and the scholar who teaches you. The page lists the differences of the four schools.