What is language?
There are so many complex aspects of language that we often don't even think of. The ones that probably come to mind first are talking (i.e., language production) and understanding (i.e., language comprehension). Language also encompasses our abilities to read, write, assign meaning to things (i.e., semantics), and communicate without words. Even thinking to yourself is related to language system activity.
Although researchers are understanding more each day about how the brain processes language, some of the main players are listed here. Language regions in the left hemisphere are heavily relied on in people who are right-handed, but their duties are somewhat more evenly split between both hemispheres in people who are left-handed.
Key Language Brain Regions
Broca's and Wernicke's areas are definitely the celebrities of the language system. They get most of the attention because they play important roles in language production (by mouth) and comprehension (by ear).
These celebrities have support staff that sometimes help with their language jobs, but also work with other clients (i.e., brain systems) doing different types of jobs. For example, the auditory cortex is important for processing information that you hear. When it's working with other language regions, this region outputs information from what you're hearing to Wernicke's area for auditory comprehension.
Angular and supramarginal gyri are complex regions referred to as "multimodal association areas." This means they receive auditory, visual, and somatosensory inputs and contribute to all kinds of sensory processing. In language, they are key in putting together different sounds that you hear or symbols that you see (i.e., phonology) to recognize them as words, as well as know the meaning attached to those words (i.e., semantics).
 Chai, L. R., Mattar, M. G., Blank, I. A., Fedorenko, E., & Bassett, D. S. (2016). Functional network dynamics of the language system. Cerebral Cortex, 26(11), 4148-4159.
 Kroll, J. F., Bobb, S. C., & Hoshino, N. (2014). Two languages in mind: Bilingualism as a tool to investigate language, cognition, and the brain. Current Directions in Psychological Science, 23(3), 159-163.