This is the 10118th most frequent Russian word.
бедным
"Бедным" means "poor" (describing someone lacking wealth or resources).
The word "бедным" is used as a dative plural adjective in the sentence "Он помог бедным бездомным людям" to describe and modify the noun "людям" and agrees in number, gender, and case with it.
Он помог бедным бездомным людям.
He helped the poor homeless people.
In "Он был бедным, но всегда счастливым," the word "бедным" is an instrumental singular adjective describing the subject "он" and indicating a transient state.
Он был бедным, но всегда счастливым.
He was poor but always happy.
Here in the sentence "Будучи бедным студентом, он экономил деньги," "бедным" functions as an instrumental singular adjective related to "студентом," forming a circumstantial participial clause.