The word is the name of a Japanese folk art movement, which came into being in the 1920s Japan. It is also a museum dedicated to this notion.
1) The Idea behind Mingei is that it is hand-crafted art by anonymous people who are not formally trained in fine art. Another criteria is that it has to be used by the masses in their daily lives and be culturally representative for the region in which it was produced. The founder of the movement Yanagi Sōetsu, has on the one hand done a valuable job in recording and raising the awareness of oriental folk art. On the other hand, he has been criticised for making the movement a nationalistic display of Japanese superiority over e.g. Korean- and Western folk art.
2) The Mingei International Museum was founded in 1974 and opened in 1978. It is has a far wider international focus, and is situated in Balboa Park, San Diego in California.
You can read more about art movements here