The word is originally just the Arabic and traditional word for "school". And, as the Wikipedia article on the topic points out, not even the usual word for a university-level institution (the traditional word for that being "Jamia").

However, the "Madrassas of Pakistan" have been in the news lately. (See this blog entry, for example.)

Reproduced below is part of a list from a Wikipedia article. Please add information as we go along. We can move as much of the material as appropriate back to the Wikipedia later: