Realms are just another name for "servers" that you can use on this game. When you initially start, you are automatically put in the newest realm. You can play on as many realms as you want per account.
Realms are "closed" after they reach a certain number of players, and are not accessable to players unless they started a city before it closed. New realms are opened as the newest realms become full, which usually takes about a week or two. Older realms are re-opened at times to allow new players to join them, this is the only time you can join a closed realm though.
The best way to start this game is to get on a new server ASAP when it opens so you will not be behind the other players. This doesn't mean much to you when you first start the game because you are just figuring out how to play... but if you want to be a major competitor, you should make a new realm as soon as you get a decent grasp on things on the 1st day it is open, preferably less than 6 hours old.
When you start a new realm, you will begin at 0 power, the same as when you initally began playing. No items, buildings, researches, resources, or troops can be carried over to the new realm.
Rubies bought on the account do transfer.
If you start on one realm and come back to find your city completely emptied out and back to 0 power, you may have accidentally changed realms. To go back to your original realm, follow the photos below.