On the more mundane side than homelessness and poverty (note, when I say “you”, I am not saying that Dopers have these):
A common one among non-horsepeople is that ponies are baby horses. They are not; foals are baby horses or ponies and have long, spindly legs with little bodies and short necks. Ponies are either proportioned like normal horses or have short, thick legs with stout bodies. Think of it like dog breeds - a Chihuahua is not a baby Great Dane.
Another misconception about animals - they do not, in fact, understand English as a matter of course. They are very good at reading body language, tone of voice, and emotion, and they can learn words, but giving your puppy a 10-minute lecture on chewing the carpet while you were gone does not, in fact, make him understand, feel guilty, and want to apologize to the carpet. It may, however, make him think that you are a nutter who has a thing against chewed-up carpets, and next time he will go hide when you come home to a chewed-up carpet because he doesn’t want to sit through another diatribe in a foreign language.
Further, when your animal or even your little kid is terrified of something, whimpering/crying/shaking/etc., petting/comforting them does not teach them to BE scared. In fact, if you associate a positive emotional state - from being comforted or even rewarded - with the scary thing, they may be less scared next time. It’s OK to expect them to not bite/punch/run off, since those things are not OK under any conditions, but distracting their attention with something positive is not going to hurt and may help. See: desensitization and counter-conditioning.
Finally, there’s the misconception that vets love animals and therefore should help them for free. Um, yes, in fact, many vets do donate their time and money to help animals because they love them, but if they didn’t get paid, they would not be able to do so and also: pay off their student loans from vet school, buy the drugs to treat your animal, buy the equipment to diagnose your animal, pay the staff that helps them take care of your animal, support themselves without having to get another job (which would interfere with the whole helping animals thing), etc. I doubt Dopers think this way, but a number of pet-owners do and complain to the vet’s face that they shouldn’t be charging anything to fix little Muffin.