Realms is a service offered by Mojang (the makers of Minecraft) to make it super quick and easy to set up your own private multiplayer games without needing to know anything about servers. Mojang does the hosting and back ups, you control who can play.
How does it work?
Once a player has subscribed to Realms (for about €10 or $13 a month), they can invite up to 20 friends to play with them on that world. The Realms that you run or have been invited to can be accessed through the main menu of the game.
Who is it for?
Realms is aimed at families and small groups of friends who’d like to play together away from the big public servers, but can’t use LAN or don’t have the ability to run their own servers.
What are the pros of using Realms?
No messing around with IP addresses or server configuration
You can be hosting a multiplayer game in an hour
You control who plays in the Realm
Only the person who buys the Realm subscription has to pay for it, it’s free for other players to join
What are the cons of using Realms?
No mods or texture packs
Currently only available for PC/Mac (coming to Pocket Edition later)
Only works with the most current version
Another possibly big downside is the fact that anyone can send you an invite to join their Realms server, they just need to know your player username. Invites appear when you log in to Minecraft, and there's no parental controls to prevent kids from accepting the invite. This theoretically makes it easier for kids to get access to servers run by people that you don't know or haven't approved. So just be aware of this aspect of Realms. Make sure to monitor the invites that your kids are getting, and teach them not to accept invitations from people they don't know.
I've had a play around with Realms and it worked really well for us, it was quick and straight forward to set up. So if you're looking for an easy way to set up your own server or want a safer way for your kids to play multiplayer with people you've approved, then Minecraft Realms is worth a try.
You can get more information and sign up at Minecraft Realms.