A mass-market paperback is the small, paperback edition of a book. They are usually around 17 X 10 CM, or about 7 X 4 inches. They are much cheaper than either the hardback or trade paperback edition, and are usually of much poorer quality.
Most serious readers of fiction have hundreds of these. They're cheap, and many books never come out in any other form. In my experience, most people will just call mass-market paperbacks 'paperbacks'. If you work around books, you will hear the term used often, as it is useful to distinguish the (relatively) standardized size of the ubiquitous mass-market paperbacks from the chaos of other paperback editions (trade paperbacks and oversized paperbacks are produced in every size under the sun).