There are special evolving buildings that each host a specific 'pet' - an animal that provides you with benefits and also provides you with expiring, temporary benefits each time it is fed 'pet food'. Here is a basic list of them, and their properties:
Let's start the feeding list:
Fire Phoenix 24 hours - 10-50% attack bonus
Coldfire Phoenix 24 hours - 5-14 KP per started scout
Storm Phoenix 3.5 hours - increases magical manufactoring up to 100%
Aureate Phoenix 3.5 hours - increased standard boosted goods production up to 300%
Ashen Phoenix for duration of a special event - gives a few % increase in 'event currency'
Twilight Phoenix 12 hours - resurrects 1-5% of TOTAL troops engaged after battle.*
Astral Phoenix 12 hours - discounts cost of diplomacy in the Spire from 2% to 10%.
Moon Bear 24 hours - +3 KP per neighborly help chest
Brown Bear 12 hours - 10-50% barracks only training speed.
Polar Bear 5 days - 10-20% tournament cool-down reduction
Panda Bear - 1 hour per ANY pet being fed (?) - increased T1 production
Red Panda Master - 72 hours - adds up to 3 vision vapor per crafted magic academy item.
Watchful Winter Owl - 24 hours - adds random boosted standard goods to Neighborly Help Chests.
Tinlug, the Star Serpent - 60 hours - gives 3 Combining Catalysts when 5-10 tourney provinces are completed, depends on stage.
In addition, each building can give a combination of goods, coins, supplies, squads, culture, population, etc. on a daily basis without being fed, but must be harvested.
*Best used with a high level Polar Bear or you will burn through your pet food rapidly.