Creatures can be frozen from their 'Manage' page.
Frozen creatures no longer gain clicks, which means they will never progress to the next stage in their growth. Eggs will not hatch, hatchlings won't gender, and gendered hatchlings will not grow into adults. As such, frozen creatures can never breed.
Because some players prefer their creatures to be frozen at the minimum amount of clicks required to reach that stage, the Rejuvenating Fairy item has been created.