It is somewhat vulgar and I wouldn't use it in front of my grandmother.
Anything you can follow 'talk' with, you can follow 'bitch' (in the verb sense) with:
That guy is always bitching about his job
Don't come bitching to me about your job
And so on...
In reference to your edit:
A teacher certainly wouldn't use it, and neither would a responsible parent (of younger children - teenagers perhaps). While it's relatively mild, it is still something of an expletive.
In short, you wouldn't use it in any polite conversation. You would only use it a situation where you would otherwise be comfortable cursing/swearing.