This is a VERY ESSENTIAL QUESTION for which every new chicken owner needs to find their answer long before they may ever need to personally apply the information. Too many folks wait until they have an injured or ailing bird before they even begin to think about this, meanwhile the poor animal is continuing to suffer because of its owner's neglect.
There is a very wide range of attitudes people have towards their chickens and purposes for keeping them. From pampered pet to producing livestock and all variations in-between. There is also a lot of variation in the amount of resources people are willing or able to use in caring for their chickens. Some have professional veterinary care that is available and affordable for them, others must contrive to tend by themselves to all their birds' emergencies. Also, some folks are willing and able to take the extra steps needed to care for a misfortunate bird that is handicapped or deformed, others keep chickens for a particular breeding program and will routinely cull chicks if they don't match their criteria.
It's all good, but what's even more essential is to know just where YOU stand when it comes to YOUR birds.
I have learned how to quickly & humanely dispatch both adult chickens & baby chicks. For me, I see no need to further stress my birds by driving them off to a vet's office, no need to incur that expense for something I can easily do myself. For adult chickens I use cervical dislocation. I either hold the bird under one arm and yank its neck out & bend the head up to break its neck, or hold the head down with a broomstick over the neck & yank up on the legs. With young chicks I cut their heads off with sharp kitchen shears. I wrap the chick in a paper towel with its neck exposed, hold it over a hole I've pre-dug near a bush, and cut quickly through the neck, and let both parts of the chick fall into the hole.
I cull when the bird is beyond help, when I've tried everything I can think of to cure it and it's still suffering. And also if it's so badly injured it has no hope of recovery. If it were a bird that turned out to be the wrong sex or have the wrong appearance I would at least raise it for its meat, or give it to someone else. And it sometimes helps to have a dispassionate 3rd party to do the job for you, they can focus more on doing a good job & less on the emotional aspect of the event.