Just when you thought you couldn't get more jealous of the witches and wizards in Harry Potter — the apparating! Quidditch! The robes! — you realize that they also get to have an especially cool relationship with animals.
Reddit user throwitway22334 pointed out that since Parseltongues can speak to snakes, maybe some witches and wizards can speak to other animals. Whether there's a specific language (like Parseltongue) is unclear, but there's definitely evidence that witches and wizards can communicate more effectively with animals than most muggles can.
For example, as Powerful_Artist pointed out on the Reddit thread, Dumbledore gives orders to his phoenix Fawkes, who also understands how to save Harry with his tears in Chamber of Secrets. And all of the witches and wizards are able to get owls to deliver their mail. Carrier pigeons are a thing in our boring old world, yes — but carrier owls? Hedwig even sacrifices herself to save Harry, suggesting a closer-than-average human-owl bond.
Even people who grew up in magical families but don't have powers themselves (referred to derogatively as "squibs") seem to be able to form close connections with animals. For example, Hogwarts caretaker Filch has an almost telepathic link to his cat Mrs. Norris, and as NoxWild says on Reddit, Arabella Figg sends her cat Mr. Tibbles to make sure Mundungus Fletcher is properly guarding Harry — and he reports back to her when the wizard slips off. Plus, Hagrid has a close bond with Fang, his enormous boarhound, who is emotionally intelligent enough to offer consolatory licks, even if he is a bloody coward.