There are two ways of canceling an appointment.
You can cancel it directly from the health card of the pet by going to the specific appointment, press the Actions button - Edit Details - Cancel appointment.
Another method to cancel an appointment is from the calendar, by clicking on the appointment card, then the edit button (pencil icon), then the Cancel Appointment button.
Canceled appointments don't get deleted. You can set whether you want to make them visible or not in the calendar by going to Settings, on the bottom left hand-side of the screen and checking the Show Cancelled Appointments box.
Canceled appointments are also visible in the health card of the pet by using the Canceled filter under the Appointments section.