The Book: No such scene occurs. Nothing in the text indicates Galadriel is capable of teleporting. Gandalf leaves with the dwarves, and they don't leave without telling anyone.
What difference does it make? This scene helps to humanize Gandalf a little bit, by showing a more vulnerable side of him as he opens up to Galadriel. Furthermore, it provides the audience with a bit of his reasoning that is not included in the novel. Gandalf's separation from Thorin's Company now necessitates some invention on Peter Jackson's part, instead of just following the novel's plot and keeping them together until the goblins' ambush.
My Opinion: I really do not like this scene. This is another example of a little change that is entirely unnecessary. It doesn't really add anything that the movie needs, and just interrupts the pacing. The movie's already three hours long, we don't need five minutes of Gandalf and Galadriel talking about their feelings. Speaking of which, what the hell was up with that? This scene jumped things up from affectionate, deeply respectful friends to full-on romantic tension.
For god's sake, Galadriel, you're a married woman!
I also really hate Gandalf's little speech about how the "little acts of kindness and love, not great power, keep the darkness at bay." It sounded deep and poetic, with that lovely music playing and Sir Ian McKellan's wistful tone of voice, but, c'mon. Are you going to use small acts of love to fight the Balrog, Gandalf? How about when Sauron's army comes marching on Minas Tirith - will you hope that the small folks will defeat several thousand orcs by being nice to each other? Small acts of kindness are great, don't get me wrong, but they're not going to do shit against a fucking dragon.
Providing Gandalf with a reason for not leaving with the dwarves (staying behind to participate in a meeting with the White Council), instead of having him narrowly evade capture at the hands of the goblins, is a good decision, I think. It's not that big of a change, and it provides a somewhat more plausible reason for their separation.
Also: Galadriel can teleport now? Are we just gonna make up super powers as we go along, like this is a Silver Age Superman comic?