Effects are changes to the overall appearance of a creatu, and tend to have a certain theme such as "Frostlight", "Undead" "Stardust" or "Cosmic". These are the only effects currently in game-play. Effects can be toggled on or off via a pet's module on your user profile. They are generally obtained through holiday events (ie. Trick or treating during Halloween had a chance of turning your active pet undead), or via use of specific items (Curse of the undead, cosmic salve, stardust urns and frostlight icicles). 

To see whether a creatu is available in a certain effect you can to go the creatu species list via the information tab at the top right of the page. If an effect is listed in the drop-down box when viewing a creatu, you can see the art available.If the effect is not listed for the creatu it is not currently available in that effect.

Alternatively, there's a helpful user-created reference here;