A mug is a type of heavy-duty cup, usually made of ceramic and usually made with a single handle. It is most often used for drinking coffee or hot cocoa, tea, or soup.
Mugs are also a weird cultural phenomenon in America, and, as far as I am aware, across the developed western world. Mugs are cheap, sturdy, and easy to customize with colors, words, and images, and perhaps most importantly, they seem useful. However, these factors have conspired to make them some of the most over-produced items in the world. Most people have a collection of mugs (whereas the optimal number of mugs to own is not much more than one per person), and there are shelves of them at thrift stores, herds of them at yardsales, a miscellany of them in staff rooms, and pockets of them anywhere else theoretically useful things can be abandoned. They are used as cheap (but useful!)
prizes,
gifts, employee appreciation tokens, packaging for candy, and for advertising for your favorite hobby, personality trait, or saying.
At the time of this writing, you can buy a bog-standard mug printed with the words or image of your choice for somewhere between $2.50 and $12.50, depending on quality. However, if you buy an order of 10,000 or more you can get the price down to $0.38 a mug, so it's a great deal if you want to give out 10,000 of something... and plenty of organizations do.