The most powerful one is phonics -which teaches the sounds of words; the second, less influential but still prevalent, method is whole language -which concentrates on stories (the stories others have written and the stories the children themselves write about their own experiences). Of the two, phonics yields somewhat better results. Even at its best, however, it does not make a significant dent in the reading failure rate.
The problem is not that the skills taught via phonics and whole language are unimportant. They are vital. The problem is that they are not enough. Each system concentrates on only one skill when effective reading and writing requires six skills. So even when children are trained in both systems, they are missing four of the six key domains needed for effective literacy. With a foundation so riddled with holes, success is inevitably problematic.