Mut'ah marriage is an Islamic marriage which have many common things with the permanent marriage (Nikah) and few differences. Mut'ah marriage has certain conditions which be followed e.g, it is not allowed to do Mut'ah marriage with a woman who is currently married. It is also not allowed to do Mut'ah marriage with a woman who was married to another man and she is still in her waiting period Iddah.
Mut'ah marriage is not allowed with non Muslim woman except People of Book after permission from you Muslim wife according to many Maraaji'.
'Wassalam.