We use -s after names, as in English, but usually without apostrophe.
It's uncommon to use this kind of possessives with other nouns than proper names. In those cases we rather use 'van', although -s is sometimes possible in colloquial language.
So, to use your examples:
Janes auto = Janes car
De auto van de vrouw = the woman's car (lit: the car of the woman)
See also "Turning a proper name into a possessive noun
" at the grammar section of this website.