If you pay for something, you are a client. If you pay to get your dog groomed, you are a client of the dog groomer (and so is your pooch).
If you shop at a certain store or use the services of a particular business, then you are a client of that store or business. If you want to go into acting, for example, you'll need to become a client of a talent agent. In computer terms, a client is a computer that makes a request of another computer, called a server. The term client is especially used in networks where one server may handle multiple requests from multiple clients.