The idea of popular education (often described as "education for critical consciousness") as a teaching methodology came from a Brazilian educator and writer named Paulo Freire, who was writing in the context of literacy education for poor and politically disempowered people in his country. It's different from formal education (in schools, for example) and informal education (learning by living) in that it is a process which aims to empower people who feel marginalized socially and politically to take control of their own learning and to effect social change.
Popular education is a collective effort in which a high degree of participation is expected from everybody. Teachers and learners aren't two distinct groups; rather, everyone teaches and everyone learns! Learners should be able to make decisions about what they are learning, and how the learning process takes place. A facilitator is needed to make sure that new ideas arise, progress, and don't get repetitive, but this isn't at all the same thing as a teacher. In popular education, then, we can't teach another person, but we can facilitate another's learning and help each other as we learn.
In popular education, the learning process starts with identifying and describing everyone's own personal experience, and that knowledge is built upon through various activities done in groups. After the activity, a debriefing process allows us to analyse our situation together; seeing links between our own experience and historical and global processes in order to get the "big picture". Through the generation of this new knowledge, we're able to reflect more profoundly about ourselves and how we fit into the world. This new understanding of society is a preparation to actively work towards social change. In fact, in popular education, the education process isn't considered to be complete without action on what is learned; whether it be on a personal or political level.