It's the immaterial part of a person that contains their personality and thoughts. But wait you say, don't we have a brain for that? Yes and no. By using a computer analogy, the brain would be like the hardware (CPU, RAM, etc.) and the soul would be the operating system. All humans have a brain composed of roughly the same things, just like most of our other organs are composed roughly the same.
The reason I believe animals do not have souls is because they do not understand right from wrong. They simply act on instincts. They are hungry, the kill for food. They need to reproduce so they find the first female/male that is willing to let them. Souls also separate us from the animals. Would you like to be compared to animal? An animal that shits itself and sleeps in it because it doesn't know better? An animal that gets locked in a cage every night at the zoo? I personally wouldn't.
We humans have instincts but by in large, we overwrite them because we have morals, which could be argued to part of souls. We are simply "programmed" different from animals.