to ask someone if they would like to have something or if they would like you to do something:
[ + two objects ] I feel bad that I didn't offer them any food/offer any food to them.
She was offered a job in Paris.
Can I offer you (= would you like) a drink?
"Would you sell me that painting?" "What are you offering (= what will you pay) for it?"
[ + to infinitive ] My father offered to take us to the airport.
[ + speech ] "I'll do the cooking," he offered.
to say a prayer or make a sacrifice (= an act of killing or giving up something) to a god:
Dear Lord, we offer up our prayers...