The 'auxilary drive belt' in most cars connects several things - it's usually driven off the bottom of the engine, goes round alternator, power steering, air con, maybe water pump (it varies from car to car). From what you've described, it sounds (to me) like something had seized, so wouldn't turn. That's why the belt got hot and eventually broke. You could try turning pulleys by hand to check they freely turn (you won't be able to turn the big one on the engine!).
Everything should be 'easy' to fit - but access is usually difficult unless you're very flexible!
Shan't be anything serious. Basically fan belt and pulley related