What is the Islamic and legal status of mut'a in Sunni and Shia schools of thought? What are the conditions for it means the circumstances in which it is halal and haram. Please also describe the details about mut'a.
First let's start with the definition.
Temporary Marriage: is a marriage between a man and woman in for a predetermined period of time.
According to the above definition, temporary marriage is not allowed in Islam according to the Sunni View for any case.
Please refer to the references for more details about the conditions of Marriage and why temporary marriage is forbidden.
Mut'ah has been mentioned in the verse 24 of the Qur’an. No body doubts it was halal (lawful) at the time of the holy prophet(s); it was lawful at the time of Abu-Bakr and a part time of 'Umar. Then the latter forbade it. Later, some narrators fabricated hadith supporting his conduct that contradicted the holy prophet(s).
Hence it is allowed Provided all conditions for it are fulfilled. conditions are also given on the referenced page (i am not writing them here because it was not asked).