Aw man. I always love questions like this! Cx I'm boring though, so whatever.
Whenever I think of making a character, it's like I've been trying to remember someone I knew a long time ago. I try to think of things they would do, or foods they like, or of memories I have from somewhere. Or, if I like a character from some story, but one aspect of theirs bothers me, I always tweak it, and then apply that personality to some other character I've thought of somewhere. Most of the time, though, I just decide to throw things in the mixing pot, and then pick out everything I don't like. That's really all I do, anyways...
Anywhere, everywhere. Imagination, anime, real life, other books...list goes on and on.
For my most recent story, the idea itself started with my main character. I was getting back into drawing and she popped onto the paper randomly. Whenever I tried to draw another person, she was there. I tried to draw items (ice cream, umbrella, bridge, etc) but she still came out of my head and onto paper (eating the ice cream, using the umbrella, dancing on a bridge, etc). So I soon got curious about who she was. Over a week I learned more about her and eventually, her name. She introduced me to two more characters while I introduced her to two of mine. And thus a story started to form.
Since I have many contradicting aspects to my personality, it's very easy to base characters on myself and still make them different from each other. Generally speaking I tailor them to situations where I think they can work best.
Hmm...my characters are either a representation of my subconscious or based off of the personalities of my friends or people I have seen, even only in passing. My dreaming mind does most of the plotwork, so I dunno where it's pulling the characters from, really.
If the characters have a lot of significance, and maybe aren't the main character, then I'll name them after a character in another story or in mythology (things that aren't copyrighted of course) to relate them to a certain set of ideas.
I design characters around aiding the plot line and mainly use them as symbolism for moral lessons. Besides that, everything else comes and I augment ideas until I feel the are decent enough to be apart of a story. Very odd, I believe - but effective for myself.
The "Characters need to work with the plot" answers are spot on, but I sort of feel like that's a given. A villain could be in it for the money or simply misled, and either one of those options could work with the plot, so there is some freedom to build characters outside of the basic details the plot demands.
Personally, I don't think I'm "inspired" so much as think of a situation or combination of characteristics that I believe would be interesting. For example, one of my characters is a genetically engineered bodyguard who's been conditioned to protect one specific person. To begin with, she follows the fairly rigid rules she's been given. However, she's gradually forced to make more of her own decisions as she realises that following her original instructions to the letter would be disastrous in some situations. The result is a character who wants to do what should logically be the best thing, but finds this often feels wrong.
There's more to the character, but I think that particular detail is the most important thing. Ordinarily, a character would more likely be tempted to do the wrong thing for practical reasons (money, their own safety, etc.), so having someone whose own sense of morality was their biggest flaw seemed interesting to me. It also seemed like a neat twist on those Spock characters who ordinarily value logic over emotion.
Usually mine are based off of the plot, or maybe a certain way I already wanted the character to be before even starting the story. I sometimes add in things that are like me, (e.g maybe the character is shy, or a bookworm), sometimes I'll get inspiration from other people I know, or maybe sometimes how they look determines what kind of personality they have.
My characters are based off of my plot, like many others have said. They are created to fill the positions needed to make the plot work. As for their design and whatnot (how I see them), that comes more I suppose from their personalities and names and what I think of when those traits are mashed together... if that makes the slightest bit of sense.
I think my method is similar to most, just not the same as most of the answers you've received so far. Once in a great while there might be some specific inspiration, but what I do much more often is model a character on someone specific, or an amalgam of real people, based on what the story needs.
Genre doesn't have anything to do with it, I think, but I write mostly fantasy.