Adaptations come in many forms—painting to song, song to poem, poem to picture--but the ones we are most familiar with are adaptations of written literature to film, be it for television or movies. Many still seem to believe that an adaptation should be a simple visual illustration of the text and that any change to character, event, or plot is an affront to the integrity of source material (ASoIaF fans, I'm lookin' at you).
But adaptation is not illustration; it is much more akin to translation—you begin with one language (the novel) and must translate it into another (film). Writing and film are indeed separate languages, with functionalities, tropes, devices, strategies, tricks, and tools, some of which translate well into the new language and others for which no translation is possible.
One of the first screenwriting projects I undertook—back when I was in film school--was an adaptation of a short story to a short script (30 pages). I contacted the author and gained permission, then set about attempting to render this story I loved so dearly ("The Flowers of Aulit Prison" by Nancy Kress) into an incredibly compact space.
First things first, you must decide what it is about your source that you most wish to express—what is the core of it that moves you so? Then you must scour your source for its meat, for its broad strokes and details, for all the pieces it can yield to help you relay what you must. Many of these things will be written in ways that film will not recognize, so this is where you get inventive.
The one comment I recall regarding Kress' reaction to my amateur adaptation was her surprise at how little dialogue I used. Indeed, there was far more in her story, and her story is not heavy with conversation. I might have been in the throes of the "show, not tell" convention so hammered into us, though more likely I recognized it was the atmosphere of this story I loved so much, the tension between worlds suffered by my hero, the anguish of reality lost.
In any case, economy is, of course, a primary concern for most film adaptations: how do you say what needs to be said in half/a third/a tenth the space? This is part of the translation process—you're adapting something from a language with tons of words for everything to another language that has very few. Form of expression is another problem: how do you show something necessary that, in its initial form, is told in a manner that one cannot show? Internal thoughts come to mind, along with brief explanations of how the world works or expressions of abstract concept.
Trouble also comes when your text takes a turn that would be intolerable to watch on screen, though it may work perfectly well on the page. You want very much to stay true to your source, to honor and cherish it, in sickness and in health, but you know that to depict everything as-is would be rather akin to running the story through one of those translation programs used by illicit DVD wholesalers from Hong Kong, i.e., laughably painful.
The challenge is to create new events that end up in the same place and/or convey the same meaning that are visually and emotionally compelling. Purists will never enjoy these scenes, but such is the status of the perpetually dissatisfied.
What, ultimately, the adaptor must do is to attain fluency in both of his languages and make them sing. This is the magic, the part one cannot teach or explain. It is also what true translators do, working the double entendres from one tongue to another, rendering with passion, coaxing out the secrets, until the song of one echoes that of the other, even if, in the end, they stand apart.