If you create a myTeamSpeak account and log in for the first time with the sync feature enabled, all of your identities, bookmarks, whisper lists, hotkeys and other items that can be synchronized will be moved to the cloud. As such you will need to be logged in to access these items.
If you disable the "Stay logged in on this computer" option, the next time you start the client you will not be logged in, and thus unable to access your cloud stored items. To get the items back you will need to log back into your account. This can be done in Tools -> Options -> myTeamSpeak.
If you disable the "Synchronization" option then you will lose access to your cloud items upon applying the changes, because the client will no longer access the cloud storage. To gain access to these items again you will have to enable to "Synchronization" option again.
If you have ever been logged in to your account in the client and had synchronization enabled (which is the default option) and can no longer see your items, then make sure that you are logged in to your account and that synchronization is enabled in Tools -> Options -> myTeamSpeak.
If you want to remove your items from the cloud and only keep them locally again, you will need to do so manually by going to the respective management dialog of the item and dragging them from the cloud to the local area.
For Bookmarks, go to Bookmarks -> Manage Bookmarks -> Go Advanced and drag the bookmarks you do not want to be stored in the cloud to the lower (Local) area.
Please note that any such item will only be available on the device you're doing this on. Any other device connected to your account will no longer have the items you removed from the cloud on any other device.