Ok, I'm gunna take a crack at this. I'm not a chemist by any means... just a biologist stuck in an organic lab.
whereas in m-Nitrophenol, the resonance structures work out so that the electrons can't "get to" those oxygens and (this is just what I'm thinking) are limited to the hydrocarbon ring on m-Nitrophenol.
I think you're pretty much on spot on with what you're thinking. Since the electrons in m-Nitrophenol can't resonate to the oxygens in the electron-withdrawing group (NO
2), they're left mostly within the benzene ring. Conversely, in p-Nitrophenol they can, and thus the electron density is localized on the oxygens in the nitro group. (Thus, electron-withdrawing.)
To answer your question, acid strength (at least in the case of phenols) is directly related to the substituents. You can think of it in two ways:
First, inductively. Electron-withdrawing groups like nitro pull electrons out of the benzene ring, and thus away from the oxygen in the hydroxy group. This causes the hydrogen to be more easily donated as it has a "weaker" bond and this, of course, raises its acidity.
You can also look at it in the resonance of the phenoxide ion, like you did.
Take this situation one step further, and analyze o-Nitrophenol. It falls between meta and para. It, like para, is able to resonate the electrons to the nitro oxygens. But in an inductive analysis, you can see the electron density is closer to the O-H hydroxy bond, thus "holding" the hydrogen better.