Animals respond to visualization and empathy. Once you have a solid connection with them, look into their eyes and visualize and really feel what you desire to communicate to them. You can speak as well, but you don't always have to. Some animals are more responsive than others, it really depends on how much they care about what you want to relay. So don't be disappointed if they just walk away. Pets are the best to practice with because they already have a bond.
Cats and dogs do understand many words we say. They recognize the meanings of them when you associate them with actions or objects repetitively and consistently. For instance, with my dog, I can say "let's go" and he's up, wagging his tail, and ready to come with me. That's because everytime I'm going to take him out, I say "let's go". They're not stupid, they do learn our language. I talk to my dog all of the time and he understands a lot. Tone of voice really matters too. Dogs heighten the pitch of their bark when they're playful and lower it when they're serious. That's why dog people raise the pitch of their voice when they're talking to their dog, a lot like when we speak to children. They respond happily because they interpret it as positive attention.
Cats are fickle and don't always respond because of their nature as being very independent. It doesn't mean they don't understand, they just don't always show interest. There are some studies on the language of cats however. Like when a cat blinks at you super slowly, its a sign of affection. It's like their version of saying "I love you". I used to do it to my cat and he would do it back. It's fun and sweet. My daughter mimicks cat meows and has had what seems full conversations with them! The cat meows back excitedly. I don't know what they're talking about (and my daughter doesn't either) but the cats love her for it! She talks to them in their "meow" pitch and they respond to her like she is speaking cat! It's cute.
Do some research on the animal you wish to communicate with. The psychology of animals really comes in handy along with being able to control the emotions you feel when communicating, your body language, and your thoughts. Ever wonder why animals always know when they're going to the vet? They're reading you! They do it all the time, you just need to be aware of what you put out.
And one of the most important things with animals is, treat them with the respect you would another human. Animals react better to those who don't treat them like they're inferior.