The title refers to the staircase in the Thoor Ballylee castle which Yeats had purchased and lived in with his family for some time. Yeats saw the castle as a vital connection to the aristocratic Irish past which he admired. The phrase "winding stair" is used in the book's third and most famous poem, "A Dialogue of Self and Soul."
Though this volume includes more poems than The Tower, it contains fewer famous ones. The most well-known and frequently anthologized by far are "A Dialogue of Self and Soul" and "Byzantium."
"A Dialogue of Self and Soul" depicts two aspects of Yeats' personality in confrontation. His soul rejects mundane concerns in favor of metaphysical contemplation, while his self (which sits with an ancient Japanese sword on its lap) cherishes worldly concerns and affirms the sufferings of Yeats' life. Self is given the final word.
"Byzantium" is a sequel to "Sailing to Byzantium," (from The Tower), meant to better explain the ideas of the earlier poem.