When you first install the app, but have not yet logged in to ZombieLink from within the app, your copy of the app is not linked to any ZombieLink account. Any runs you do while this is the case are stored on your phone, but are not backed up to a ZombieLink account.
Once you log in to ZombieLink from your copy of the app, it is linked to your ZombieLink account. Any runs stored on your phone are now uploaded to ZombieLink and are backed up to your ZombieLink account.
If you log out of ZombieLink from your app after your app has been linked to your ZombieLink account, you will still be able to see all of your run logs and your base, however any changes to this data will not be synchronised to your ZombieLink account.
Please Note: A ZombieLink account is the only way to save/backup your data. We do not automatically back up your account details to the Cloud, or create a save file on your phone. If you want to make sure your Zombies, Run! history is saved we'd strongly recommend creating a (free) ZombieLink account.