Of peripheral interest is the fact that I can now look back on my own language-learning experiences and classify them into these methods, along with all the value judgments associated with them (some methods are very much frowned upon these days).
A quick primer:
The Grammar-Translation Method, aka The Classical Method. Language students sit in class and discuss L2 (the second language) almost exclusively in L1 (the native or common language). They read from L2 texts and translate them into L1. The written word is paramount and you can get out of a Grammar Translation class without ever having learned how to actually speak. The focus is on conjugation drills and the minutiae of grammar. This is how we learn dead languages, or languages we only need to do academic research.
The Direct Method, aka The Berlitz Method. I have a soft spot in my heart for this one. Basically some educators realized that sometimes it would be nice to, you know, communicate in a foreign language you've learned, and they came up with this approach. Berlitz has taken it mainstream. Check out their commercials here and here and here. Students of this method use only L2 in class and have to rely on the teacher's pantomime and use of real-life objects to convey the meaning of what is being taught. The Direct Method tries to skip over L1 and directly activate L2.
The Audiolingual Method, aka The Army Method. In the 1950s and 1960s, two things happened. First, Skinner's Behaviorist model of human learning, with all its repetition and imitation, became huge. Second, the United States decided it needed people to spy on the Russians and put military personnel through foreign language training using this method. The Audiolingual Method featured the teacher as the conductor of the class, and students repeated and mimicked and parroted canned phrases and dialogues until they could spit out gems like "I want a chicken salad sandwich" like a native. The problem is, we've since decided that that's not the best way to learn. Also, students of the Audiolingual Method are sometimes incapable of reorganizing their memorized, fluent-sounding phrases into more meaningful language, such as "I want a smoked turkey sandwich." They're stuck with chicken salad FOREVER.
Fast forward through The Silent Way, Community Language Learning, Desuggestopedia, and Total Physical Reponse.
The Communicative Approach. This is the current darling of the field of second language teaching. It values authentic language learning situations and contexts, and aims to prepare learners to function in real foreign-language situations. The Communicative Approach recognizes that there is more to language than just grammar, or just canned key phrases, and takes into account wider issues such as pragmatics and sociocultural considerations.
I'm pretty sure I learned Spanish and German via the Communicative Approach. The German textbook we used was called Kontakte, for crying out loud. I learned Russian in Russia and Arabic in Arabia, outside of a regular language classroom, so I can't say that instruction fell under any one method.
The biggest "what the heck?" moment for me when it comes to these methods is realizing, years after the fact, that I learned Japanese at the BYU using the horribly maligned Audiolingual Method. I have no idea how this managed to happen. I have dim memories of professors defending the textbook we used (Japanese: The Spoken Language) at the beginning of each semester, but at the time, I didn't realize why they were so defensive about it. Now, reflecting back on the language lab and the memorized dialogues, it's all clear: we were using an out-of-date method.
Still, I can't really pass judgment on the efficacy of the Audiolingual Method (or lack thereof) because my foundation of learning Japanese was four years in high school, where we used some kind of hybrid method created by the teacher. He seemed to draw from all areas of these methods and from all kinds of materials and I learned Japanese from him pretty dang well.
If you have studied a foreign language, what method did your teacher use, and how did it work for you?