Red Hat is also a common but inaccurate Western name for one or more school of
Tibetan Buddhism. It may refer to the oldest form,
Nyingma or "Ancient"; or it may refer (I gather) to all of the Nyingma,
Sakya, and
Kagyu schools.
It is opposed to Yellow Hat Buddhism, more correctly known as Gelug, the later reforming school to which the Dalai Lamas belong. Although still common in some Western texts, the terms Red Hat and Yellow Hat are now deprecated in favour of the correct Tibetan terms.
Thanks to Jinmyo for drawing my attention to my misuse.