The word arena originally meant "place of combat," and it's sometimes still used this way. Its root is harena, a kind of sand that was supposedly used on the floor during ancient Roman battles to soak up spilled blood.
An arena originally was an open, sandy place. The Romans used the term to describe the site for gladiatorial combat or an enclosed area in which some public entertainment was staged. The seating for spectators was around the arena. In common usage today the difference between stadium and arena is negligible.