A person who makes coffee is referred to as a barista. The word comes from the Italian word for bartender, and it refers to the fact that the person making the coffee is usually behind a bar. In Italy, a barista may serve either alcoholic or non-alcoholic drinks.
The word barista is typically only applied to people who work in coffee houses where espresso-based drinks like lattes and cappuccinos are served. People who make drip coffee in traditional restaurants or diners usually do not have a special title, and those coffee-making people can be called by their traditional titles like server, chef or manager.