From the Minecraft Realms menu in Minecraft, you can view a list of Realms that are available to you. Each Realm that you join first requires an invitation, indicated by the flashing mail icon. After you accept it, that player's Realm will be available to you. Make sure that you have the latest public release of the game.
A Realm's current status is indicated by the colored dot next to it: green means that the Realm is open and available to join, yellow shows that it expires soon, and red indicates that the Realm has expired or has been closed by the owner.
If you're having trouble accepting invites or joining Realms, please make sure that you are not using a modded game client and are using a resource pack that contains textures for the Realm buttons. If this is the case, switch to the standard Minecraft game client, select Options, and then Resource Packs, and then return to the default.
If you'd like to leave a Realm (leave the Realm's player list and delete it from your available Realms), from the Realms selection screen, highlight the Realm, and then select Leave Realm.