Pauvre, in the sense of “not rich”, is always placed after the noun: un homme pauvre, une famille pauvre, une ville pauvre, …
Pauvre, in the sense of “unfortunate”, is always placed before the noun: un pauvre homme, une pauvre hère, …
There is a general trend whereby some adjectives have a literal meaning when placed after the noun and a figurative meaning when placed before the noun: un grand homme (a great man) vs. un homme grand (a tall man). There is a difference between pauvre and grand however: the position of pauvre is based on its meaning, whereas grand normally comes before and is only placed after the noun in a few expressions where the figurative meaning kicks in.
“L’homme pauvre” means that this man is poor, he doesn’t have any money (there are a lot of synonyms for this case like misérable, fauché, sans le sou, etc.)
Le pauvre homme is an expression that means that you have some pity for this person (in this case you can replace pauvre with malheureux).
It’s an expression that is very often used in the current language, for example you have “mon/ma pauvre !” which means “I’m sorry for you”, or “le/la pauvre” which means “I’m sorry for him/her”.
In southern France, they also say “Pauvre !” (or “Malheur !”) as an exclamation mark meaning that you’re complaining about something (never use that expression in Paris or people will laugh at you).