A Jinchuuriki is a person who has a Bijuu sealed inside their body.
The word comes from the manga/animé series Naruto. In the story, there are nine "tailed beasts" or Bijuu. These Bijuu are inhuman masses of chakra and their powers are described in the show as "beyond human understanding and uncontrollable." However, people still tried controlling them by sealing them inside people, thus suppressing their powers and controlling them in this fashion.
Two known Jinchuuriki are Naruto from Konoha who has the Kyuubi (Nine-tailed demon fox) sealed inside him, and Gaara of the Desert from the Hidden Sand Village who has the Ichibi (One-tailed demon raccoon) sealed within himself.
Jinchuuriki will show exterior signs of their sealed demons, such as Naruto's (fake or real) "whiskers (fox)" and the thick darkness under and around Gaara's eyes (raccoon).
If a Jinchuuriki ever has their Bijuu extracted from their body, they will die.
Naruto Uzumaki - Hidden Leaf Village - Kyuubi (Nine-tailed
Gaara of the Desert - Hidden Sand Village - Ichibi (One-tailed
Having Bijuu sealed within their bodies, Naruto and Gaara are both Jinchuuriki.