A stairway is a staircase — a set of stairs going from one level to another — but the word carries a hint of size or grandness, so it turns up for the sweeping stairs in a hotel lobby or a palace rather than the little steps down to a basement. The second half, 'way', is doing real work: a stairway is felt as a road that happens to go upward, a route from here to there, which is why the word fits long, impressive, or dramatic stairs so well and why it appears in phrases like 'stairway to heaven'. American English uses 'stairway' more freely than British English does, sometimes for perfectly ordinary stairs, and on signs and building plans you will see it as plain technical vocabulary with no grandness at all. In everyday speech on both sides, most people still just say 'the stairs'. So think of 'stairway' as the word you reach for when the stairs are worth describing, or when you want that sense of a path leading somewhere.