In computing, a server is a piece of computer hardware or software (computer program) that provides functionality for other programs or devices, called "clients". This architecture is called the client–server model. Servers can provide various functionalities, often called "services", such as sharing data or resources among multiple clients, or performing computation for a client.
Homebridge is a lightweight NodeJS server you can run on your home network that emulates the iOS HomeKit API. It supports Plugins, which are community-contributed modules that provide a basic bridge from HomeKit to various 3rd-party APIs provided by manufacturers of "smart home" devices.
Since Siri supports devices added through HomeKit, this means that with Homebridge you can ask Siri to control devices that don't have any support for HomeKit at all.
A router is a networking device that forwards data packets between computer networks.
Jellyfin is the Free Software Media System that puts you in control of managing and streaming your media. There are no strings attached, no premium licenses or features, and no hidden agendas.