If you could humor me, I have some confusion about firewalls in the real world

I work in security, policy to be precise (I know, everyone's favorite), and I am confused about firewalls. My certification books always refer to firewalls in different ways, and never touch on real world examples. My google search has turned up similar points of confusion.

1. What are the actual types of firewalls? Everything I read says something different, i.e., there are 3, 4, 5, 7, etc. types of firewalls. 1st gen, 2nd gen, next gen, WAF, proxy firewall, circuit-level gateway, stateful, packet-filtering, so on an so forth.
2. In the real world, when and where would each generation be used? Home, corporate, perimeter, DMZ, applications, internal databases, etc.
3. Is there a good resource that shows, at a high level, the limitations and uses for each type of firewall?