Converse.JS - A free and open-source chat client for your website
Openfire - Openfire (formerly Wildfire) is a cross-platform instant messaging (IM) and groupchat server.
Rocket.Chat - Rocket.Chat is a Web Chat Server, developed in JavaScript, using the Meteor fullstack framework.
Prosody - Prosody is a modern flexible communications server for Jabber/XMPP written in Lua.
Lets Chat - Open source self-hosted chat app for small teams
Matrix.org - Matrix is an open standard for decentralized persistent communication over IP.