There are many ways, and probably more than we know. Here are some:
Body language: The postion of the tail, the look, the motions. Some of these can be quite complicated. For example, if two new dogs or wolves meet, and fight for dominance, when one wins, the loser will roll over and bare his throat, inviting a death-stroke, and the victor will gently take the throat in his teeth, and not bite down. This is a way of saying, "I won't bite you; I want to be your friend."
Actions: Grooming each other, as in monkeys picking nits and cats licking one another.
Scents and pheromones (hormones released into the air or left on the ground that have meaning): An endless variety of messages are carried by scents within a species and between species. Insects and even plants play this communication game.
Sounds: My cat purrs. Does yours?
There are many other modes of animal communication, and, actually, a lot of them work for people. We have pheromone scents; we say "ahh" when we enjoy a touch; we communicate emotions with body langauge and facial expression.