It's a trilogy by Philip Pullman, the first of which being "The Golden Compass". The other 2 books are called "The Subtle Knife" and "The Ember Spyglass". I recommend all three.
Anyway, TGC begins in a place called Oxford, a sort of parallel to the Oxford in England. But the inhabitants there are all born with a 'daemon'. The daemon has an animal form, but that form changes during the person's childhood, as the child changes, from moment to moment. Here's an excerpt:
"....he (the daemon) was a bat, an owl, a wildcat again, she looked all round, catching his panic, and then saw them at the same time as he did: two men running at her, one from the nearer side holding a throwing net."
So you see, the daemon sensed the danger first and got immediately into action. Basically, in this world, a person is never alone nor lonely, because one always has one's daemon, closer than the closest possible friend. You always understand your daemon and your daemon always understands you. The most painful thing that can happen to a person/daemon is to be separated from each other.
The daemon settles into one form as the person grows up. Excerpt:
"Why do daemons have to settle?" Lyra said. "I want Pantalaimon to be able to change forever. So does he."
"Ah, they always have settled, and they always will. That's part of growing up. There'll come a time when you'll be tired of his changing about, and you'll want a settled kind of form for him."
"I never will!"
"Oh, you will. You'll want to grow up like all the other girls. Anyway, there's compensations for a settled form."
"What are they?"
"Knowing what kind of person you are. Take old Belisaria. She's a seagull, and that means I'm a kind of seagull too. I am not grand and splendid nor beautiful, but I'm a tough old thing and I can survive anywhere and always find a bit of food and company. That's worth knowing, that is. And when your daemon settles, you'll know the sort of person you are."
I don't want to say too much about the story itself in case you want to read it. 
