Have you heard of the Buddha (also called the Gautama), Guru Rinpoche, or Zhabdrung Ngawang Namgyel? All these three auspicious characters in the Bhutanese history have: either killed all the demons in Bhutan, or saved Bhutan from the armies of other countries, or did auspicious deeds for many famous kings, or... Well, I guess you got the idea. Monks in Bhutan follow either of these leaders. Monks in Bhutan follow Buddhism, the teaching of Zhabdrung, Buddha and Guru.
Anyway, don't be too surprised to see a monk with his mobile phone clasped to his ear, or a monk enjoying a leisurely ride in a car (or even driving one!). Monks now don't have the serious sense of religion that I think monks back then had.
Bhutanese citizens can be very touchy about their religion, so a bit of teasing or a hot debate about religion may be enough for you to end up without any friends... or get thrown into jail.