Awesome Privacy

Probably one of the most popular, secure private messaging apps that combines strong encryption (see Signal Protocol) with a simple UI and plenty of features. It's widely used across the world, and easy-to-use, functioning similar to WhatsApp - with instant messaging, read-receipts, support for media attachments and allows for high-quality voice and video calls. It's cross-platform, open-source and totally free. Signal is recommended by Edward Snowden, and is a perfect solution for most users.

Session is a fork of Signal, however unlike Signal it does not require a mobile number (or any other personal data) to register, instead each user is identified by a public key. It is also decentralized, with servers being run by the community though Loki Net, messages are encrypted and routed through several of these nodes. All communications are E2E encrypted, and there is no meta data.

XMPP, also known as Jabber, is an open standard for decentralized messaging that has been widely used for decades. It has actually been the basis upon which WhatsApp, Facebook's Chat and Google's Talk were built, but these companies (eventually) chose to remove the interoperability with other servers. Prominent XMPP clients support OMEMO end-to-end encryption, which is based on the Double Ratchet Algorithm that is used in Signal. For more hands-on information and to register an account you can visit JoinJabber.

Matrix is a decentralized open network for secure communications, with E2E encryption with Olm and Megolm. Along with the Element client, it supports VOIP + video calling and IM + group chats. Since Matrix has an open specification and Simple pragmatic RESTful HTTP/JSON API it makes it easy to integrates with existing 3rd party IDs to authenticate and discover users, as well as to build apps on top of it.