We teach animals using conditioning. Operant conditioning is a way to train animals to do what we want using rewards. We can make them pay attention to our commands by rewarding things we like.

We can even shape behavior by rewarding things that are the start of behaviors we want. Here’s an example. If you hold food out to both sides of your body, a dog will probably look at the food. If you say their name in a way that makes them look at you, then give them a treat, they will learn to look at you when you say their name. If you call your dog and offer a treat or toy, they will probably come to you, and you give them the treat or toy. Then you can back off on how often they get a reward other than a “good dog!.” You can start off just being a few feet away with a great treat that they will come to anyway. Over time, you can gradually increase the distance and things that distract them.

Some people just expect a dog to somehow understand commands, but it takes patience and work to train an animal. If you ever call an animal, then do things they don’t like, they will learn to avoid you and your calls. If you call and call, they will learn to ignore your calls.

You may be interested in studying animal behavior. Here’s an interesting article on operant conditioning: operant conditioning

A Russian scientist named Ivan Pavlov gave us one of the first answers to your question. He had a bunch of dogs, and noticed that, when he would walk by them after being away for a long time, the dogs would drool and lick their lips as if they were just about to eat. Pavlov thought that the dogs associated, or paired, the idea of getting food with him walking by. After all, he was the one that fed them. To test this idea, he would ring a bell when feeding the dogs. After a while of doing this, he tried ringing a bell without giving them food. With just the sound of the bell, the dogs would drool and get excited for the food even though there was not food around. Palov would say that the dogs were conditioned, or they learned to pair, the sound of the bell with getting food.

Using conditioning, we can train animals to understand humans. Typically when training a dog, for instance, you can pair food with a behavior that you want. So when you call your dog’s name and she happens to approach, you can reinforce that behavior by giving her a treat. If you repeat the pairing of food with the desired behavior enough times the animal eventually will perform the behavior without food. You can even build really complicated series of behaviors one step at a time using conditioning. This is how an animal can come to learn its name and other commands.

Science magazine has an exciting article about how dogs understand us. Dogs understand some of the words we say, and they hear our tone of voice, which helps them know if we are happy with them or mad at them. So the best result for the dog is when we say familiar praise words to them in a praising voice. Here's a description of the research:

Dogs have been domesticated for many thousands of years. Humans and dogs have thus had plenty of time to develop close communication bonds for work and play. Using brain-scanning techniques, Andics et al. investigated how dogs process human speech. The left brain hemisphere in dogs responds more strongly to praise words, and a right auditory brain region distinguishes intonation. Reward regions only light up if both the word and the intonation are consistent with praise. Hence, the subtlety of dog-human communication has become hardwired in the absence of language.