If you don't want to wait for the timer to reset itself and want to hatch sooner you can purchase a Purreau's Potion in the crown shop for 400 crowns. Your hatching slot is the one with a piggle icon. You can find the count down if you go into your book to 'character' (shortcut C), then click 'timeslots' (it's the greenish tab to the right with a stopwatch symbol). Your hatching timer resets automatically after 24 hours (12 hours for members).
Hatching through the kiosk still requires us to have a hatching slot, a pet that is adult or above and sufficient gold.