When talking to someone, how do you decide when to use "vous" or "tu"?
Comment allez-vous ?
Comment vas-tu ?
Comment choisir entre vous et tu quand on s'adresse à quelqu'un ?
This is one of the most subtle and complicated points in the French language. In fact, I think you could say that it has more to do with the culture than with the language itself. It's not really something that can be easily taught on this site.
Until you have a good feel for it, you can use the following rules of thumb:
Again, these rules are very approximate and everyone will have a different opinion on this, but they should keep you from shocking anyone too badly until you develop your own style. Also remember that these are based on my experience in France; different Francophone countries have totally different standards. Tu is much more common in Québec, for example.
One last thing: remember that if your non-Frenchness is obvious from your accent, French people will normally forgive impertinent uses of tu, so don't stress out too much over this.
Edit: But Tipx's point is worth highlighting: The consequences of using "tu" where you shouldn't are worse than the contrary.
You use vous:
It is customary that the teacher, the boss, the parent in law, the elder, etc. proposes using tu the first.
You use tu :
Some people use only tu, no matter what.
In unclear situations, many people tend to avoid having to use the pronoun altogether until the issue is resolved.
Vous is used when you are talking to a person in a formal situation (like your superior), or to a stranger.
Tu is used when talking to somebody you know well enough. It is often considered okay to say tu when persons address each other using their given name.
(To address several persons vous is always used.)
Vincent gives the main things to know.
On the other hand, I am a native and I don't like to be patronized, so since I am 20 I apply a strict reciprocity rule : if you use "tu" with me I will do the same (same for "vous"). Of course, if I use "tu" with someone, I won't be offended that he do the same with me ;)
I tend to always hear "tu" in québec. And "vous" is used usually speaking formally to your boss.
You must use Vous when talking to several persons.
Using Vous when talking to only one person marks the respect, including the simple and polite respect.
You can use Tu in relaxed mode, that is :
Because of an old usage coming from the French Revolution (when the French King was removed), some rare people are used to always say Tu to everybody. This may appear very strange, even to Frenches, but it's tolerated.
When people become to be familiar, or want to speak relaxed, they often propose to say Tu : "On peut se tutoyer".
My advise would be that, when you don't know what to use, except for children (where tu is probably your best choice), would be to use vous by default as it is more respectful.
Then, at the beginning or later, if you want to be more friendly (so that's not with everyone, generally not with your boss for instance), you can ask "Est-ce que je peux te/vous tutoyer?" (which would mean "Can I use 'tu' when I speak with you?") or, if someone talking with you currently use "vous", you could tell him if you want it to use "tu", "Tu peux me tutoyer si tu veux" (You can use "tu" if you want).
I use "tu" everytime, mostly because I don't know the "vous" version of the word. E.g. tiens or tenez.
It's pretty obvious that I'm a foreigner, even with putting on a decent Bergeracois accent. My skin tone must give it away.
I've never come across anybody taking "tu" in a disrespectful way, they'd be more disappointed with just speaking English at them. Due to the amount of tourists and ex-pats who don't speak anything but English. The Dordogne residents seem to be more than happy with informal talk than being shouted at in English.
When you are talking to many persons: always vous
To your teacher: vous
Teachers to students: often tu, sometimes vous but only for teenagers (and always vous in the university)
To persons of your family: tu, even to old persons (you might see people using vous with their parents in the old books, but it is not used anymore)
To children and teenagers (from an adult and among them): tu
To people you don't know well: vous
To your boss: vous, unless (s)he tells you to say tu
And there is a very important rule that is very useful for us French and even more for foreign people: when you don't know whether you have to use tu or vous, always say vous, because it is less embarrassing to respect a person too much than not enough.
There is a simple rule of thumb based on what you would call the person in question.
Ex: use "Tu" when you are on a first-name basis, and "vous" if on a last-name basis.
Use "Vous" for people you address by
Use "Tu" for people you address by
There are many situations where not following this rule is accepted, but not mandatory.
Ex: During middle and high school, teachers may address their students using various combinations like Lastname+Tu, Firstname+Vous, etc.
To summarize, vous is the formal version of you, an example of this is the sentence Je Vous Aime, yet the informal version of this sentence is Je t'Aime.