Matrix.org is a non-profit open source project that publishes an open standard for decentralised interoperable communications (IM, VoIP, etc), an example server implementation and a bunch of example client SDKs, bridges, bots, etc. 'Matrix' is the name of the resulting ecosystem on the 'net.

Riot.im is one of the various client apps for Matrix, supporting iOS & Android, and is probably the most advanced at this point.

EDIT: clarify org structure: Matrix.org is being set up as a non-profit UK company owned by the various individuals running it. The business model is to create a new ecosystem for interoperable comms that everyone can benefit from.

Riot is a for-profit company (technically called Vector Creations Ltd) which is a subsiduary of Amdocs (a big telco supplier company). The business model is to offer paid hosted services in future for users wanting commercial-grade hosting and hosting private bots/bridges/etc.