One of the main reasons for the development of printmaking is the desire of artists to make more money from their work by selling multiple copies; printmaking satisfies this motive. Much early printmaking was used only for copying existing works of art, and was considered then as posters are considered now: a downmarket version of a great work. From the 18th century on, though, artists commonly pursued printmaking as the primary medium for some or all of their work.
Prints by artists today retain their financial value as art (i.e., as an appreciating investment) because they are created by an artistic process, rather than a strictly mechanical one, and because the number of multiples is limited. In Rembrandt's time, the limit on the size of an edition was practical: a plate degrades through use, putting an upper limit on the number of images to be struck. Plates can be reworked and restored to some degree, but it's generally not possible to create more than a thousand prints from any process except lithography. A few hundred is a more practical upper limit, and that allows for significant variation in the quality of the image.
Because of that variation in quality, lower numbered prints in an edition are sometimes favored as superior, especially with older works where the image was struck until the plate wore out. In later times, printmakers recognized the value of limiting the size of an edition, and explicitly numbering the prints (e.g., a print numbered 15/30 is the fifteenth print in an edition of 30). With printmaking earning respect as a legitimate medium in its own right, the printing of editions with tight controls on the process to limit or eliminate variation in quality has become the norm (though monotyping is a subdiscipline of printmaking that is deliberately opposite this precept). Artists may print an edition much smaller than the plate allows, keeping the edition within the undegraded lifespan of the plate; or specific steps may be taken to strengthen the plate, such as electroplating intaglio images (using an electric process to put a very thin coat of a stronger metal onto a plate of a weaker metal).
The discipline of releasing prints within a specific edition was a later invention. Rembrandt would create plates, print a few hundred copies, rework the plate, sometimes changing the image radically, and then print a few hundred more. As long as the print could make money, it was printed. Nowadays, prints are not only released in small editions, but the plate is usually cancelled afterwards: marked or destroyed to explicitly prevent any more prints from being struck. This is an expectation of collectors and investors, who want the prints they buy to retain their value.
The conventions for numbering prints are well-established, but there are other marks to indicate that the print exists outside of an edition: artist's proofs are marked "A.P." or "P/A"; monoprints and uniquely hand-altered prints are marked "unique"; prints that are gifted to someone, or are for some reason unsuitable for sale, are marked "H.C." or "H/C", meaning "hors de commerce"--not for sale. Finally, a master image may be printed, against which the members of the edition are compared for quality; these are marked "bon à tirer".