Facebook Messenger is one the most popular instant messaging application. However, it is not so secure as for an example WhatsApp or iMessage from Apple. The problem is that Facebook uses the secured connection (i.e. HTTPS protocol) but the messages are not encrypted by additional protocol. It means that everyone who have access to Facebook servers can read all messages.
WhatsApp and iMessage encrypt messages, so only the sender and recipient are able to read this.
The real encryption works in this way, that private and public keys are needed during the decryption process. The public key is exchanged between two persons, however the private key is stored on the device. It means, that you will not able to read your messages if you login to your account on different device. Of course, you can make a backup of your private key and restore it on another device, but this is a different story.
It looks that Facebook Messenger will provide end-to-end encryption.
This feature is called Secret Conversations and it is visible in the newest beta version of Facebook Messenger for Android devices. However, currently it is not possible to enable this. Very interesting is the fact, that the secret conversation will be visible on only the device that it has been activated. It means that you will not be able to read secret messages on your computer via Facebook page. Also, you can very easily delete all encrypted chats using a single click.