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.)

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:

• If you are young (say, 24 and younger), use tu with anyone your age and younger, unless the situation is very impersonal (e.g. with a cashier in a shop).
• If you are older than this, use tu with young children (in any situation) and with adolescents and young adults in social situations.
• Use tu on the Internet unless you’re trying to be very formal, or you’re talking with someone whom you know offline and address as vous.
• If you are in school or university, use vous with your teachers, no matter their age.
• If someone who is your social equal (co-worker with the same job, for example) uses tu, you can safely use tu as well. This may not apply to social superiors (but that’s open for debate: see Sylvain’s answer! In any case, it’s far less clear-cut.)
• In all other situations, use vous unless the person you’re talking to specifically asks you to use tu.
• (May seem obvious, but needs to be included): Always use vous when addressing more than one person. This is a grammatical rule and you will not be understood if you use tu.

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.

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 😉

In France

Vous use cases

You use vous:

• by default, when you don’t know the person and the person is older,
• by default, when you don’t know the person and the person has the same age and you are over 30 years old,
• as a mark of respect to someone (for example a father/mother in law),
• when you want to sound old-fashionly romantic (saying vous to a woman that you generally address with tu can have a certain charm when used properly, as a mark of a mix of respect, tenderness, shyness, and more),
• by default, when addressing a customer, or more generally someone you are in relation with through professional obligations (but not part of the same organization as you).

It is customary that the teacher, the boss, the parent in law, the elder, etc. proposes using tu the first.

Tu use cases

You use tu :

• when you are well acquainted with the person,
• when addressing children / teenagers (some teachers might prefer to use vous with teenagers though, to build some distance and respect with the pupils),
• when you are well acquainted with the person, and not in a public situation where you have to show some distance with that person (well acquainted politicians and journalists use tu only off the record, for example),
• within left-wing political parties, the norm is to address one another with tu (the more to the left the stronger),
• when you want to sound young and cool (radio broadcast animated by people who target a young audience will systematically use tu when on the air).

Remarks

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.

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 :

• When you talk to a young person (when they not seems adult yet, lets say about 18
  to 24). Note that you have to use Vous for hight titled young
  person, like a young prince)
• Very young kid that don’t know to speak well can say Tu (lets say until about 4 or 6 years old).
• When you talk to people you are familiar with. That is your family, your friends, some of your neighbors, may be some closed colleagues at work, … You can say Tu to God. Note that in some families of the hight society, child have to use Vous to their parents.
• When you talk in associative collaboration where it seems to be the usage (associations, Internet
  forums, even at work in some companies, …)

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”.

• 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.

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.

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.

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Use “Vous” for people you address by

• Last name (Smith, Dupont)
• Title (Miss, Sir, Professor)
• Honorific (Your Honor)
• Profession (President)

Use “Tu” for people you address by

• First name (Abigail, Robert)
• Nickname (Abby, B-dog)
• Family title (Mom, Granpa)

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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.

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.