Dogs chase cats because dogs by nature are natural predators and act upon survival instincts to chase and catch smaller creatures to eat. The survival instincts can be modified by training or a pet behaviour specialist. The natural instinct that encourages dogs to chase cats is triggered by movement.
When you throw a toy to your dog, the dog rushes to chase, catch, and throw the toy about in its mouth in order to “kill” its prey. For this reason, dogs chase cats because the cat is moving around the house and in doing so, triggers the dog’s natural instinct to chase the cat.
Some cats actually will not run from a dog and indeed the cat will take a stance and start hissing and spitting at the approaching dog. In most cases the Dog will stop in it’s tracks and go no further.
Some medium to large-size dogs can potentially kill a cat if they do catch them but most well-fed dogs chase cats for the fun of it.