The icon will display on boarding screens to indicate a pet who does not have a previous active boarding appointment in PetExec. You may be seeing this icon for pets who have boarded before if you are new to PetExec and that pet does has no boarding history in PetExec. In that case, it will also serve as an indicator of first time you are managing those pets boarding in PetExec. It is also possible for "first time boarder" icons to appear for some pets in the same family and not others. This can happen if there is a new pet that has been added to an owner, and that PET has not boarded before, but his/her siblings do have a boarding history.
The first time boarding icon can be disabled by going to menu-->company preferences-->application preferences-->boarding category-->Enable First time boarding Icon. Uncheck the preference and click "Update Preferences" at the bottom of the screen.