This is a section I'm not really sure what to do with. I'm aware that there exists this idea within Islam of temporary marriages. However, it is not contained within the Qur'an. The references to it that I have seen are all ahadith, and we know how I feel about them. I don't believe that the idea of mut'a is very widely accepted in Islam. The idea, though, is that men can marry women for a predetermined length of time. Three days, at the least. And then they divorce, and there's no sin on either of them.
From what I can gather it was something that was allowed, pre-Islam, and then Mohammed forbade it after the battle of Khaibar. I'm just going to reproduce the hadith referenced and the Qur'anic verse that the author says is used to support it and leave it at that.
sahih Bukhari 7.62.13o - " Narrated 'Abdullah: We used to participate in the holy battles led by Allah's Apostle and we had nothing (no wives) with us. So we said, "Shall we get ourselves castrated?" He forbade us that and then allowed us to marry women with a temporary contract and recited to us: -- 'O you who believe ! Make not unlawful the good things which Allah has made lawful for you, but commit no transgression.' (5.87)"
sahih Bukhari 7.62.52 - " Narrated Jabir bin 'Abdullah and Salama bin Al-Akwa': While we were in an army, Allah's Apostle came to us and said, "You have been allowed to do the Mut'a (marriage), so do it." Salama bin Al-Akwa' said: Allah's Apostle's said, "If a man and a woman agree (to marry temporarily), their marriage should last for three nights, and if they like to continue, they can do so; and if they want to separate, they can do so." I do not know whether that was only for us or for all the people in general. Abu Abdullah (Al-Bukhari) said: 'Ali made it clear that the Prophet said, "The Mut'a marriage has been cancelled (made unlawful).""
sahih Bukhari 9.86.91 - " Narrated Muhammad bin 'Ali: 'Ali was told that Ibn 'Abbas did not see any harm in the Mut'a marriage. 'Ali said, "Allah's Apostle forbade the Mut'a marriage on the Day of the battle of Khaibar and he forbade the eating of donkey's meat." Some people said, "If one, by a tricky way, marries temporarily, his marriage is illegal." Others said, "The marriage is valid but its condition is illegal.""
And the verse from the Qur'an surah al-Ma'idah 5:87 - " O you who have believed, do not prohibit the good things which Allah has made lawful to you and do not transgress. Indeed, Allah does not like transgressors."
So from what I can see, mut'a marriage was made impermissible, but those who want it to be allowed say that if Allah said it was okay, then it has to still be okay. Only I don't see where Allah ever said it was okay. There's no mention of it in the Qur'an, and if it's not in the Qur'an, it's not from Allah. Or at least that's the theory I'm working with.