to give or receive hospitality: to host and care for a guest, or to be taken in and lodged as one.
The word comes from the Greek word for a stranger or guest (ξένος), and it names the whole exchange of hospitality that word set in motion. In the ancient world, taking in a traveler was a sacred duty, not a small courtesy: a household was bound to shelter, feed, and protect the guest, and such a bond could even be passed down for generations. So the word carries more weight than the English "to put someone up for the night."
Which side of the exchange it describes depends on the sentence. Sometimes it describes the host who welcomes and provides for a visitor; sometimes it describes the guest who is taken in and lodged. Either way the relationship is the same one of hospitality.
In Acts, Luke uses the word to trace how the good news crossed from Jew to Gentile: Peter, at first a guest lodging in one house, becomes a host to Gentile messengers, and much later a pagan official on Malta shelters Paul's shipwrecked party. Hebrews turns the same word toward every reader, recalling how some who took in strangers were unknowingly hosting angels.