A dairy is a business that uses animal milk. A dairy may be the part of a dairy farm where the milk is collected, pasteurised and made perhaps into butter or cheese. Sometimes there are separate places for milking cows (the milking parlour) and processing the milk.
In the past, before refrigeration and quick movement of milk by rail from farm to city, cows were kept in the city and milked right next to where the milk was to be sold. The dairy, was then the shop that sold the milk, butter and so on.