This would be serial storytelling, which is quite commonly found on television these days, but originated in literature, though many of the most famous "serial" books are typically sold with the complete story intact, making it the literature equivelent to binge watching a season of television on Netflix.
Originally, the format was used by many authors to publish their books much quicker than publishing full length novels. Typically the author would sell the story to a literary magazine which would publish the next chapter in each issue and would typically publish the whole story over the course of a preset negotiated issue period (typically weekly or monthly). Others had in house writers who would either work on the next chapter of the same story or an anthology style. As I said, many a famous classical series were published in this way including "Uncle Tom's Cabin", many of Charles Dickens' works ("The Pickwick Papers" being the novel that established this style's viability), "Sherlock Holmes" (most were self contained adventures of the characters, but Hound of the Baskerville was a "multipart episode" equivalent for the time. The death of Sherlock Holmes was actually Doyle's attempt to leave the series despite the high fan demand for the works.
Serialization did make the jump to other mediums, including Comics, Film, Radio, Television. During the golden age and to a lesser extent, even today, many comic book characters started off as one off stories in anthology titles that gained popularity to such a degree, they came to dominate the title's story output. Action Comics debuted with it's signature character, Superman, and these days it's a title in D.C.'s Superman comic lines. A similar situation exists for Batman and Detective comics, but in some cases the Anthology comic predated it's signature hero. For example, Amazing Fantasy was 15 issues deep before it's star attraction, a friendly neigborhood hero named Spider-man debuted... in what was to be the final issue before the entire magazine ceased production, and Thor first appeared in the 83rd issue of his featuring anthology Journey into Mystery. These days, the "Anthology" title of a major superhero typically retains a second story that will feature a member of the feature hero's cast or a promotional story line for an upcoming new title comic series. Further more comics these days tend to run a serial storyline with each episode being a chapter that will last from 2 to 6 issues and then collected and sold as a "trade" book where the whole story is told.
Cliffhangers originated in golden age Film serials as a gimmick to ensure the audience comes back for the next entry. Typically, a Film serial was intended for kids and featured many popular Superheroes (Superman, Batman, and Captain Marvel (the one that yells "SHAZAM" not the one that Nick Fury has on Beeper) are some classic examples, but Flash Gordon had several series). These were normally cheaper than a theater's evening attractions, which is saying something when the evening shows were a dime, and that included the film, a news reel, a cartoon, and change for popcorn. Even then, day shows often did double features so that could be two films, a news reel, a cartoon, change for your popcorn and three hours for mom and dad to not have to worry about Jr. Cliffhanger endings were staples (so named because they typically had the heroes hanging from a cliff, though the danger where they would break the story did change... but the unresolved situation would hang over the kids head for a full week before he could find out that they were saved within a minute of the show's start). The film genre died as TV came to take over serial formats, though it would see a revival by George Lucas in the form of Star Wars and later Indiana Jones which was a love letter to pulp serials of the day, and later evolved to it's final form, the cinematic universe, where superheroes once again became the top dog.
The genre died when episodic storytelling became much easier on TV, and the promise of finding out more when you "TUNE IN NEXT WEEK" was even cheaper as the shows rarely had a multi-episode story-line. I Love Lucy did several gimmick, the most famous of which was the show's portrayal of Lucile Ball's real life pregnancy. By a lucky chance, the birth of Lucy's TV son aired mere hours before the birth of her real life son (the episode was taped in advance and the script decided to say the TV child was a boy, despite not knowing the real child. The papers did joke that "Lucy sticks to script" the next morning.
Serialized TV grew slowly, but it became much more prominant with the advent of DVD collections of a show's season (over coming the difficulties of people not being able to reliably see a critical episode of a season). These shows still split the episode into mostly "Monster of the week" and "Mytharch shows" with the former dominating the bulk of a single season while the later would often be built close to planned production breaks (Typically Mytharchs would be in the first episode of a season, a cliffhanger surrounding the winter break, and a season finale (Either eliminating the season long story or cliffhanging into the next season). An occasional mytharch story was thrown in the midway point between the mid-season break and the season's opening and closing episode. The rest was pretty much filler (Mytharch and Monster of the week originate from an early user, X-Files, with Mytharch relying on the alien conspiracy while the Monster of the Week was anything but the aliens and rarely connected to anything else, with a few twists thrown in... a few mytharcs started as feeling like monster of the week (Red Museum), while some Monster of the Weeks were deliberate spoofs of the alien plot (Joe Chung's From Outer Space).
Serial literature has been making a comeback however, thanks to the internet and blogs, an author can self publish a story on his/her blog and do a serial installment (typically weekly) and some discussion of notes or behind the scenes or AMA engagements with the audience occupying the wait time between story content. The novel/web novel "Worm" about a superhero/villain is one of the noted early successes of this style with the author having concluded the novel and published it in book format and having produced two other novels in such a format and presently working on a Worm sequel as well. The limitation here is that much of the advertising needs to be self generated, facilitating online engagement from the author that wasn't required from the Magazines of yore.