I know that Godzilla was always made of latex. The first suit was made crudely and quite dangerous (but it is memorable!). The later suits were relatively safer, but were still quite heavy, hot, and stifling. Three minutes were all you could stand. The mouths moved from radio controlled mechanisms inside the head (the actor wears a brace around his head for it), and the tail was manipulated by overhead wire operators.
Since Godzilla Vs. Biollante, the dorsal fins were made seperately, so they could be attached to the back. The skin texture was made into the mold (before that, they pasted on urethane chips for that bumpy texture on the old suits). Since Godzilla 2000: Millennium, the principal suit has an oxygen hose attached to the tail for the actor to breathe.
Of course, there were different ways to bring the monsters to life. From puppets to animatronics, for the more lifelike features. Even CGI became useful!
The Ultra Monsters were pretty much done the same way, only lighter to allow more flexibility for the actors inside. They were still quite a burden on the stuntmen wearing them!
Monsters from Toei Superhero shows go without saying. They are made with all sorts of materials, mostly latex.