Yeats, like a ton of poets, was pretty preoccupied with women. While Yeats was sort of a ladies' man, that didn't start until his 30s or so. Though he loved and was close with many MANY women, Maud Gonne, an Irish nationalist and actress and all around badass, was his main romantic focus for literally decades. He first met Maud in 1889, and, immediately infatuated both with her beauty and her political leanings, he began developing a lifelong love for her. Many of his poems are about her, ranging from praising to disparaging to despairing. In this way, she is his Muse, the source of focus for his creative energy.
However, Maud isn't a demure and dainty Muse figure. She is full of fire and bite. She is not a wilting flower. She had a long affair with Lucien Millevoye, a French journalist, which resulted in the birth of her daughter Iseult (who I'll get to in a bit). She was English by birth, but got really into the idea of Irish independence and national culture after living in Dublin.
"The world should thank me for not marrying you."
She herself knew how infatuated Yeats was with her, and that he considered her his Muse. She repeatedly rejected his proposals of marriage, replying that the art he created with her inspiration made him happy, even when he protested his misery at being without her. And, let's be real here, she was probably right. Yeats was prolific and a wonderful writer, and having a Muse figure to keep him sexually frustrated uh, inspired? was a good thing for him.
Maud also completely subverted patriarchal conceptions of Muses, considering herself not the mother but the Father of Yeats' poetry. Really cool, right?
She married John MacBride, an Irish nationalist, in 1903. Yeats was really pouty about it, to put it likely. The marriage fell apart fairly quickly, and they separated. Maud retained custody of their child, Sean.
Aside: Now, in 1916, Yeats (still single) proposed to Maud AGAIN, and was rejected AGAIN. But then he notices that Iseult, now 22, is pretty gorgeous, and proposes to her too after "a summer of wooing", whatever that means. (He was rejected.)
The Muse trope in poetry is common. Dante's Beatrice and Petrarch's Laura are just two examples. But the Muse is often a passive figure, an object of adoration, seen from afar and not necessarily conversed with. Yeats and Maud were genuinely very close, corresponding for decades even when they were apart and often working together on building up the Irish national culture they both valued. The reason Maud is so valuable as a Muse is because of her activity---she was vivacious and passionate and Yeats knew her very well.
Overall, Maud was ridiculously independent, talented, and intelligent, which is ultimately what captivated Yeats and gave her the title of his Muse.