The Salem Witches' Institute is a magical institute, possibly a school, presumably located in Salem, Massachusetts. Based on its name, it presumably only accepts female members. Some of the institute's members attended the 1994Quidditch World Cup, and were gossiping happily in between two tents bearing their school's name.
Behind the scenes
Some fans believe this to be the American wizarding school that J. K. Rowling claimed in an interview would be mentioned in Harry Potter and the Goblet of Fire. However, the same book makes mention of another "American" school, the Brazilian wizarding school, presumably in South America. Additionally, the only known members of this institution are described as middle-aged (though they could be teachers or alumni).
Assuming it is not a school, the institution may be a magical counterpart to the Women's Institutes, though these primarily exist in Great Britain and Canada in the real world, and muggles would not actually know if there was a magical women's institute in the United States because it would have all kinds of muggle repellents on it.
The banner which displays the school's name is described in Harry Potter and the Goblet of Fire as a "spangled banner", possibly the American Flag, which is "spangled" with stars and referred to in the American National Anthem as "the star-spangled banner".
If it is indeed a school, the name "Witches' Institute" would suggest it is a girls only school, probably indicating that there is indeed a boys school too.
It is also possible that the name is merely derived from the Salem Witch Trials, and may in fact have nothing to do with the gender of the students that attend it.
It is assumed that this school or institute is based in Salem, Massachusetts, the setting for the Salem Witch Trials.