Populaire, when it refers to the level of language, means that it is typically used by lower social classes.

Argot denotes a vocabulary used originally by the underworld (in the sense of thieves, prostitutes, etc.). It can be used by other social groups but will still be rated negatively as language used by “bad boys/girls”.

Familier means it is a level of language used typically among interlocutors sharing a high level of intimacy (family members, friends), but not in eg. the press, academic writing, professional contexts, etc.

Note that if you compare similar entries in multiple dictionaries, you will see that the border between familier and populaire is rather thin and that a same word can be rated either as familier or popular in different sources (eg the word tronche is rated as populaire by the TLFi and as familier by the Wiktionary).

In French like in English, the formality of a word, or lack of, can and often do evolve during the time. Its popularity too. The TLFi sometimes considers informal what is mainstream today.

In any case, while it is useful to know the various ways to say something in French, it is better to stick the the standard way until you are very familiar with the language level of the variants and you know exactly what levels are expected during a conversation.

A typical word English speaking people have often an issue translating to French is “guy“.

Here are various ways to translate it:

• Un individu : formal
• Une personne : standard
• Un monsieur : often familier, child language (m’sieu)
• Un mec : used to be argot, is now familier
• Un gonze : argot, outdated
• Un type : familier
• Un keum : argot (verlan)

Very often, I have heard foreigners using type or mec (because they have guy which is colloquial in mind) in situations where native speakers won’t have used these words but just personne.

I guess this is somewhat similar to a non native English speaker writing I wanna in an e-mail.

To the sound definition of the three terms given in this answer I’ll add the following appreciations. First, whether “familier” (informel), “populaire” or “argot” the three registers carry with them a certain degree of vulgarity; ranking in increasing degree of vulgarity we find “familier”, “populaire”, “argot”. Second, one must keep in mind that “argot”, if sometimes translated as “slang”, does not correspond most of the time to English slang (as this answer clearly allows to make out); “argot” is of a register comparatively more vulgar than “slang”, except for what is called “coarse slang” and it is comparatively much less acceptable in French than English slang is in English.