I'm still learning about this topic, but in the meantime, i invite you to visit this resource that I found that sums it up pretty well:

• Direct language method - this is basically immersion in the language being learned and it focuses more on pronunciation and less on grammar
• Grammar-translation language teaching - teaching through translation, so using both languages and educating your child on what it means. Grammar and vocabulary is developed through memorization, and there is less focus on verbal communication. It's the most common language teaching method when learned in secondary school
• Audiolingual language teaching - This heavy focuses on hearing and speaking, and less on written
• Structural language learning approach - For those who like rules, this method is for you. Also often used it secondary school, it's also the most common learning approach in language learning apps and sees language as a complex of grammatical rules, like verb conjugation.
• Total physical response (TPR) - In this teaching method, your child physically does what you tell them to do. You could ask them to read a book or come to the table to eat. This method stresses aural comprehension and kinesthetic learning.
• Communicative language teaching (CLT) - with this method, you can teach your child based on situations. For example, you can make believe she's at a supermarket and needs to buy some fruits and vegetables.
• Task-based language learning - This teaching method is "requiring" the learning of a language to complete a task. It builds on what your child already knows and highlights what they need to know to complete a task. Note- there is little error correction in this language teaching method.
• The natural approach - this language teaching approach stresses the similarities between learning the first and second languages. There is no correction of mistakes. Learning takes place by the students being exposed to language that is comprehensible or made comprehensible to them.

Northwest Career College lists out the various language teaching methods