Open the relevant Customer Record and click 'Pets'. Press the 'Edit' button next to the name of the pet you want to delete to open the pet's record. Scroll to the bottom and use the red "Delete this Pet" button.
Please Note: if the pet has any bookings in the system, you cannot delete it unless you delete those bookings first. Alternatively, you can mark the pet as deceased / inactive so it no longer appears in the pet list.