The Language Centers refer to areas in the brain that are responsible for the functions of understanding and producing language. Traditionally, the Language Centers in the brain refer to Broca's area and Wernicke's area.
Broca's area is that part of the brain which when damaged, renders a person unable to express himself through speech, even when he can fully understand language. Wernicke's area on the other hand, is that part of the brain involved in understanding language. When this part is damaged by a lesion, a person's speech becomes senseless and incoherent.
Although both areas are still considered to be important in language, modern brain imaging techniques that allow us to see which parts of the brain are active when processing and producing speech have shown that the two areas are just part of a wider network that is involved in language.