While re-imaginings of stories have been told over and over in film and other media since long before film began, the first instance of the use of the term "reboot" was 'The Incredible Hulk' in 2008, which came out only five years after the previous movie 'Hulk'.
The studio felt that they needed a word to help describe what they were doing with the movie because it was not a sequel, nor were they starting over with another origin story. Instead, 'The Incredible Hulk' started with a quick gloss over the origins of the Hulk and then jumped ahead four years, leaving it open-ended as to if the events from the previous movie were considered to have happened or not in this story arc (as we now know, they did not).
It's also worth noting that there were questions at the time as to why the movie was being remade so close to the previous film, and no one could call a movie with a full recast a "sequel." This put Marvel in a position where they needed to create a marketing spin, and thus the term "reboot" was born. At this time, they didn't want to tip their hand to the idea that they were going to start a series of Marvel movies that all had plot tie-ins to each other, nor had they constructed any long term plan for such a series (the series we now know as the Marvel Cinematic Universe, beginning with 2008's Iron Man).
As pointed out above, the term was needed because of the proximity of the two movies. In 2005 when Christopher Nolan's new vision of Batman surfaced with 'Batman Begins', it was referred to generally as a "re-imagining of the Batman universe." It has since been retroactively deemed a "reboot," but the time gap as well as the complete change of tone between Nolan's Batman series and the 1989-1997 Batman series makes it unnecessary, particularly when considering how long ago the first series began.
Ultimately, the term has given us a much needed word for movies that fall into a certain category, we just never knew we needed the word until 2008. The word has since gained a more concrete definition: to discard all continuity in an established series in order to recreate its characters, timeline and backstory from the beginning. It's overuse in the comic book genre has generated some push back from fans when the term gets retroactively applied to media older than the term itself, which may also have to do with the vague and ever-changing definition.