When Amazon has a book listed as "paperback" it's generally the same size as the hardback edition it just has a paper cover instead of pasteboard. They are what are also known as "first runs" meaning they are part of the initial sells and marketing phase. Independent authors like myself will often not have the choice of hard covers so all of our books will be sold as "paperback". Some well-known authors will have an initial run in hardback, followed by a "paperback" edition which usually costs a little less than a hardback.
"Mass market paperbacks" are the ones that are smaller (length and width, not necessarily thickness). These are what traditional publishing houses release after the initial sells and marketing run. When traditionally published authors have sold a set number of books in hardback or "paperback", the publisher will release the "mass market paperback" which will usually be about half the price of the hardback edition.