Original comment by RafaelCLP
This article is very good, but I want to demystify some things. I would feel more comfortable knowing why I'm addressing someone in a certain way, than just knowing that I should because that's how it's done in that language. I'm not sure if this is useful for someone else, so I can only hope this helps someone.
"vós" is, indeed, almost dead. You probably won't hear it anywhere, except from some older people in Portugal. But it's still taught in schools. If you use it, people will find it weird, but will definitely understand. It's been replaced by "vocês" (plural form of "você").
"você" has its origin in "Vossa Mercê" (vossa mercê - vossemecê - vosmecê - você), which was just an honorific to refer to the king (similar to His/Her Majesty, or "Vossa Majestade", in Portuguese), and quickly fell into vulgarity. And that is why, when using "você(s)", the verb conjugates in the third person, instead of the second person. That applies to all titles/honorifics (despite "você" is not exactly one, it has its origins in one and the language treats it the same way). Referring to someone as "você" is not polite. When talking to an older person or someone of superior social status, you would use "O senhor"/"A senhora"; when referring to a doctor you would use "O doutor"/"A doutora" or, even more politely, "O senhor doutor"; when talking to a judge (mainly in court, of course), you would use "Vossa Excelência" (same as "Your Honour/Honor"); and so on. Learning how to properly and politely address everyone in every occasion is hard, since you need to know all the proper honorifics, but you can at least learn how to politely address almost everyone (as you do in English) - we will get there after the examples.
Here are some examples: (A) "Doutor, eu estou doente?" (Doctor, am I sick?) (B) "O doutor acha que estou doente?" (Do you [doctor] think I'm sick? -- the direct translation would be "Does the doctor think I'm sick?", but it doesn't make sense in English when "the doctor" is the person you are talking to; in Portuguese it's fine!)
(A) "Senhor, poderia me informar como chegar ao mercado por favor?" (Sir, could you please tell me how to get to the market?") (B) "O senhor poderia me informar como chegar ao mercado por favor?" (Could you [sir] please tell me how to get to the market?)
(A) "Vossa Excelência, trago documentos que comprovam o fato." (Your Honor, I bring you documents that confirm the fact.) (A) more formal (and correct): "Excelentíssimo senhor juiz, trago documentos que comprovam o fato." (B) "Vossa Excelência exige mais provas?" (Does Your Honor require more proof?)
Notice the difference between (A) when directly addressing the person you are talking to ("O senhor ...") vs (B) when calling that person ("Senhor, ..."). "Vosso(a)" is a possessive pronoun (same as "Your") and doesn't call for "O/A" before it (although you could use it).
Being polite is just a matter of pleasing the person you are talking to without sounding rude to them and/or those around them/you. To most people, it's okay to just use "você", and I use it almost every time. However, if you want to make sure you are not being rude, it's very common to just use "O senhor"/"A senhora". It's as polite as being addressed as "Sir" or "Ma'am" in English. Of course you wouldn't use it to address a judge in court, but except for those very special occasions that call for a more appropriate honorific, it's polite enough (and sometimes even too polite...). Addressing a stranger in the streets when you want to ask for information; addressing your professor; politely addressing your parents; in all these situations, and in many others, it would be polite enough.
Now, about "você" vs "tu". In Brazil, "tu" is almost gone, but it's a big country and it depends on where you live. I can only talk about where I live. I still hear a lot of "tu", but they do not conjugate it appropriately. Instead of "tu gostas" they would say "tu gosta" (as in "você gosta"), and they'd find it very weird if you conjugated it properly. I do not recommend that you do the same just to fit in. In these places, just stick to "você". I can't say much about Portugal (I'm Portuguese, but moved to Brazil 12 years ago). All I know is that 12 years ago you wouldn't hear people using "você", only "tu"; nowadays, you hear a lot of both. I was told that you use "tu" when talking to people you have some intimacy with (friends, family, ...), and "você" with others, but I can't confirm that.