An important idea in Matrix is Interoperability. This means that Matrix is open to exchanging data and messages with other platforms using an Open Standard. We refer to the connection to other platforms as bridging.
For a thorough examination of the different methods of bridging, and a discussion of the terminology involved, check out How do I bridge thee? Let me count the ways...
Portal rooms: these control chunks of room aliases namespace. For example,
#freenode_#channelname:matrix.org corresponds to #channelname on Freenode. In this way, Matrix users can transparently join IRC channels on Freenode. Portal rooms are typically managed by the remote network's side of the room.
Plumbed rooms: these rooms are "plumbed" into one or more specific remote rooms by configuring a bridge (which can be run by anyone). For instance, #matrix:matrix.org is plumbed into #matrix on Freenode, matrixdotorg/#matrix on Slack, etc. Access control for Matrix users is necessarily managed by the Matrix side of the room. This is useful for using Matrix to link together different communities.
Bridgebot-style: in this case, messages in either direction are conveyed by a bot residing on the given platform. This is a sub-optimal experience because metadata is lost. For example, all messages might be sent by the same bot, but with the message text prefixed with the name of the original sender.
Puppeting: solves the problems of Bot-based bridging by "puppeting", meaning controlling, a user on the other side of the bridge. This means that to native users, they see messages as being sent from the correct sender. Double-puppeting means this is done in both directions of the bridge. This is the most preferred way of implementing a Matrix bridge.