What is "Realms Alpha"?

Minecraft Realms is a new paid service that allows you to quickly create worlds for you and your friends. Currently only available to anyone that attended MineCon and Swedish users.

Effectively they are dedicated servers, run by Mojang.

From the Realms FAQ:

Q: I can already see the Realms button on Minecraft: Pocket Edition. What’s up with that?

A: That’s there for selected testers. You’ll be able to host your own server soon!

https://mojang.com/2013/12/the-minecraft-realms-beta-has-launched-in-sweden/

https://minecraft.net/realms/faq