A livery is a place that will take care of your horse, for a fee. You will mostly see this use now in historic novels — especially the kind where a cowboy rides into town and stables his horse at the livery.
The noun livery also refers to a uniform sometimes worn by male servants, like doormen, footmen, and chauffeurs. Your cousin works for a prominent and wealthy family, and he is required to wear the livery of that family when he is on duty. He isn't allowed to say where he works, but he calls his boss, "Your Highness."