The term arose in the 3e era, but the actual style of play goes way back.
Essentially, the 15-minute adventuring day is when the party expends many resources in a short period, usually only a single encounter or two, and then retires until tomorrow.
Often in 3e it manifested as "the wizard is out of his highest two levels of spells, let's rest". In 4e it's "I'm out of healing surges." In Ye Olden Days of 1e and such, it was "that was my last cure light wounds."
It's often considered a problem on a couple of levels. It encourages pcs not to hold any resources back and to 'nova' every encounter; it often leads to wussy play (dude, really? Nobody is even wounded but you're knocking off for the day because Fizlor is out of 6th level spells?).
The problem is largely alleviated by random encounters, proactive enemies, attacks while you're trying to rest, etc. The dm doesn't have to use these techniques all the time, but if you know that your rest period isn't a sacred safe time, it changes the 'go nova all the time' part of the dynamic, if nothing else.