When you hire someone, you invite them to work for you or the company you represent. If you manage your local ice cream shop, it might be part of your job to hire extra ice cream scoopers for the busy summer season.
Your boss at the animal shelter might hire you immediately if she's desperate for help — or if you impress her as a true animal lover. Once she does, she can call you "a new hire," or a recently hired employee. It's less common in the U.S. than in Britain, but you can also say you hire a car when you rent or lease one. Hire comes from the Old English hyr, which means both "wages" and "interest."