ESL traditionally was taught with a grammar intensive method. You would have textbook explanations of the grammar, and many exercises to re-enforce structure and usage. You would then have a reading exercise, and at the end of it the teacher would try to draw out some conversation. Many people find this method tedious and boring and have very little success in learning or speaking English. This was basically the only way to learn ESL until the mid 1960s when other methods were developed.
The direct method was developed as a reaction to the traditional method. Rather than starting with a book and grammar, the class begins with a conversation in idiomatic English. Visual aids such as videos and pictures are used to stimulate interest. The mother language of the students is never used for expanations. If you want to learn it, you learn it in English. The type of language used depends on the target objectives of the class. These objectives can range from street slang to highly academic usage. One of the principals of this method is that like a tail follows a horse, grammar will follow the spoken exercises.
The reading approach methodology has the student read extensively. Topics such as American history and culture are assigned. The downside of this method is that students are often nervous about speaking. The audiolingual method is based on the direct method with more emphasis on the exact repetition of words and phrases. In the learning counseling method, the person teaching the class tries to be a counselor rather than a teacher. He guides the students and reacts to problems rather than simply feeding imformation.
The total physical response method has both the student and the teacher physically doing what is talked about. To teach the word jump the teacher will jump, then have his students do the same. The community method tries to relieve a student's stress about learning and not pose threats. Finally, the natural response methodology tries to teach adults English in the same way they learned their native language -- by listening and reacting. There are many other methods and new ones are being formulated constantly. The goal of these methodologies is to have the student speak idiomatic English.
All of the methodologies have a certain validity. Different people learn English in different ways. On top of this, according to Stephen Farr, in his book "Teaching as Leadership," different cultures learn in different ways. Latins are nervous about being corrected and want cooperative learning situations. Africans want interaction. Koreans feel that asking questions is disrespectful to the instructor. These cultural differences influence the choice of a methodology.