OED 2nd ed. sez:
The selection and arrangement of separate cinematographic shots as a consecutive whole; the blending (by superimposition) of separate shots to form a single picture; the sequence or picture resulting from such a process.
I can't remember the first time I saw this device used in a film. Most often it's used to condense a long period of time filled with repetitive activity into the space of a few minutes. The activity depicted, while tedious, is necessary for the development of the plot. The boxer training for the big fight; the failing student cramming for his final; the figure skater falling again and again before finally landing that double axel; the young pianist practicing for the international competition; the miraculous new couple in various memorable city locations, eating ice cream and laughing (their words to each other obscured by the soundtrack, giving them temporary privacy). This device has become so common that it's not shocking at all to see time compressed in this way. Books have their rows of asterisks and breaks between chapters, plays have their scene changes and intermissions, but these solutions to the problem of time now seem inelegant next to the movie montage, which shows us just enough of the boring, necessary shit without utterly boring us.
In real life, the proportions are reversed; the moments of portent and revelation, the plot points our lives seem to turn on, last mere minutes or even seconds. The punch that knocks your opponent to the ground, the memory lapse in the second movement of the concerto, the hesitant intake of breath before the first kiss: all fleeting, gone before you know it. The bulk of our lives is the work and the repetition and the tedium, the same crap again and again. And this crap forms the foundation that our opinions, experiences and accomplishments are all based on.
This is what concerns me about the ubiquity of movie montages: they are teaching us to hate our lives. We want our days to have the plot and pacing of a well-constructed story, and instead we get these dull details. Rebel against the movie montage! Don't perpetuate the myths that make us miserable. Learn to pay attention to the details, the stuff you've deemed too boring to care about. Don't sit there waiting for the next Big Important Thing to happen to you. Look around. Everything is potentially interesting. This is life.