A brouter is a device that functions as both a bridge and a router. It can forward data between networks (serving as a bridge), but can also route data to individual systems within a network (serving as a router).
The main purpose of a bridge is to connect two separate networks. It simply forwards the incoming packets from one network to the next. A router, on the other hand, is more advanced since it can route packets to specific systems connected to the router. A brouter combines these two functions by routing some incoming data to the correct systems, while forwarding other data to another network. In other words, a brouter functions as a filter that lets some data into the local network, while redirecting unrecognized data to another network.
While the term "brouter" is used to describe bridge/router device, actual brouters are pretty rare. Instead, most brouters are simply routers that have been configured to also function as a bridge. This functionality can often be implemented using the router's software interface. For example, you may configure a router to only accept data from specific protocols and data sources, while forwarding other data to another network.
NOTE: Since routers are more complex than bridges, it is more likely for router than a bridge to function as a brouter. Therefore, brouters are also called bridging routers.