Minecraft server hosting as a paid service has been around for well over a decade - however up until Realms we hadn't seen such a tight connection with the game developers, instead third party companies (like ServerFlex) would provide the hosting and you would connect to a server by typing the IP address of your server into the multiplayer part of the game. With Realms you can create your own Minecraft world in the cloud and play with up to ten friends without having to be on the same network.
Realms is a complete hosting service provided by the makers of Minecraft (Mojang) who were more recently acquired by Microsoft.
The platform was thrust onto the world stage in April 2014 after a pilot was ran in Sweden in December 2013. Minecraft Realms is a service ran by Mojang that allows you to host your own server for both Java and Bedrock Edition.