I've never had very strong mental images of the character of fantasy races, short of however they're presented in any given franchise. Dwarf Fortress though has changed my mind a little, if only because I've probably gotten closer to it than any particular setting.
Humans: As a student of human history, it's hard not to think of humans as infinitely variable, but the way they appear in DF inspired me a bit. I see them as a smooth mix of British, Germanic, and Arabian styling (if only because DF humans wear turbans so often). A little bit Monty Python, but with a High Medieval religious austerity. And lots of horses and mauls and just some weird touches, like tanned fish hides. Picture a guy in platemail covered in silks, skins, and iconography, wielding a giant mallet or a whip and buckler, riding an Appaloosa.
Dwarves: Likewise, I'm informed more by DF than anything, although their limited portrayal in Tolkein's books, especially The Hobbit, to me paints a slightly different picture than the usual Always-Lawful-Stone stock idea. To me, the Three Commandments of Dwarvehood are: Do what needs doing, do it right, and don't complain about it. And all the same, they're kinda dumb and thickheaded, even in the more Steampunk portrayals. Unscrupulous, not at all friendly, utterly pragmatic, impervious to tragedy right up until they snap completely, and concerned first and always for the Dwarves.
Elves: I draw more on the Vulcans for Elves. Wise, mighty, and proud of it although they'd never admit it. They've been around forever and stuck their more-knowledgeable-than-thou fingers in every pie, like a whole Illuminati race, if they ever bothered to do anything besides observe and make dramatic last stands. But they can screw up, and screw up spectacularly when they do. They're the cause of and solution to all the world's big metaphysical problems.
Goblins, Orcs, Bugbears, Hobogoblins, etc: Basically more human than human. Humanity is the scrappy underdog of the "good" fantasy races; corruptible, numerous, and more intelligent than perceptive. The greenskins (if they be green at all, which I'm not adherent to) are all the bad parts of humanity magnified. They build ugly cities with ugly machinery and ugly animals, breed like disease, and rule underworld nations away from the sunlight. Trolls are the comic relief, and I love the old-school bridgemonger concept more than any other treatment.
There's other races too I guess, but this is more about the "standard" ones. One of my favorite parts of Planescape: Torment, a whole franchise about turning fantasy on its ear, was the little enclaves of sentient undead and cranium rats and minor demons. The Cantinas and Monster Towns are my favorite locations when they show up, because they give due justice to the "other" fantasy creatures that usually spend their short lives as narrative speedbumps, and portray them as rowdy but negotiable working stiffs, my kind of people.