Linguist here. This is from memory from a teaching class required for my major some years ago. I'll dig out my old textbook if you'd like me to cite sources.

In the US, the Grammar-Translation method was the way to learn a language. If you've taken a Latin class, you probably learned by Grammar-Translation (although some new-fangled books are coming out that take other approaches). This method is how people learned language for centuries, going back at least to Rome with Greek language. It wasn't until WWII that the Audio-Lingual and oral methods were developed by Leonard Bloomfield and Harold Palmer, respectively. But I should mention, even with the new methods being used by the government, classrooms still largely used Grammar-Translation until the 60s (and maybe 80s, I can't remember).

The Natural Method, as I learned it, is essentially a linguistic immersion in the classroom in order to facilitate "natural" learning, and the native language is banned from the classroom. This started in Germany and France in the first decade after the turn of the century.