Written by Emily Nagoski, Ph.D.
Short answer: not a lot.
Really what happens is you increase blood flow to erectile tissues. And what happens when you increase blood flow to erectile tissues? Again, not a lot, due to the non-congruence of women's physiological sexual response with their perceived/emotional sexual response.
Sexual arousal medications have AMAZINGLY HUGE placebo effects, so it's perfectly possible (between 15-40%, I'd estimate) that a woman taking Viagra would experience an increase in desire... but she'd experience that taking two aspirin if her doctor assured her it would "increase her libido." (Of course that effect is now diminished for you, because you've read this and know it's bullshit.)
We know Viagra isn't effective in increasing women's sexual desire because you can bet your bippy that Pfizer has tested it and tested it and tried to find an effect and they can't. It's been years since it was first produced and still no effect has been proven.
This is one of the drums I bang all the time: medication isn't effective, and there are behavioral, attitudinal and experiential things you can do to increase desire/arousability for women.
Can it harm a woman? Well no one knows for sure, of course. I'm gonna say globally probably not, taken rarely and at low doses, unless you have the conditions that are contraindicated for men. And all medications have the potential for side effects or health risks; because it's not approved for women, the research on these effects and risks is limited, to put it mildly.
So what happens when a woman takes Vagra? Well most likely it DOESN'T increase her sexual desire.