Generic Ancient Greek term for an assembly hall where the citizens of a democracy would meet, or for the citizen-body itself. The duality of meaning is similar to the modern American use of "Senate", which can refer either to the group of people or to the building in which it meets.
After the temple and the theater, the bouleuterion was the most common form of Greek civic architecture in the ancient period. However, unlike the other two building types, bouleuteria did not serve as models for later Roman architectural forms. This is probably because the buildings served their purpose remarkably poorly. They needed to be large, open spaces where citizens could speak and be heard, debate and vote. Their roofs, however, were supported with post and lintel architecture, and required forests of columns to stay up. All those columns then blocked the view of (and the light on) the speaker's platform.
The Romans discovered truss roofing, brought the arch up from underground (the only place the Greeks would use it), perfected the dome, and built basilicas to serve as a model for roofed interior spaces for centuries to come. The bouleuterion vanished into obscurity, an architectural dead end, a line extinct.