Aww. That puppy or kitten is so adorable.
Experiments made in Russia with foxes indicate that breeding for "tameness" - a lack of fear of humans and willingness to stay with us - also produced features we associate with domestic animals such as floppy ears, weird tails and patched coats.
Scientists call it "domestication syndrome." And it appears to have an effect in the brain. Or rather, selecting for tameness brings out a neural crest deficit - a reduction in the stem cells that eventually create the adrenal glands, making the animal less fearful. In dogs and rabbits this is also associated with floppy ears, which are caused by a cartilage deficit - something which might be harmful in the wild, but isn't a problem for our domesticated critters. (Floppy ears in horses cause major problems with socialization and on the rare occasion that they show up, horses with the defect are not bred on).
Here's a theory, though.
Do we find animals with "domestication syndrome" cute because we actually subconsciously know they're more tame and thus less dangerous? This doesn't explain how some people can see a very high degree of cuteness in wild animals such as otters or even (albeit more rarely) snakes, but for most human beings...we look for "cute" features that include floppy ears, curly tails and flat faces.
Just a thought.