Matrix is an application layer communication protocol for federated real-time communication. It provides HTTP APIs and open source reference implementations for securely distributing and persisting messages in JSON format over an open federation of servers. It can integrate with standard web services via WebRTC, facilitating browser-to-browser applications.
Element.io is a client application built by the core Matrix team. Synapse is an open-source Matrix homeserver written and maintained by the Matrix.org Foundation.