Keats uses the word "still" in this poem with two different meanings. Of the two meanings, the first one, from line 9, is the most unfamiliar to us – because it's old fashioned. In this old-fashioned meaning, "still" means "always." Thus, when Keats says "yet still stedfast, still unchangeable," what he really means is that he would like to be "always stedfast, always unchangeable." The second meaning of "still" is one we're familiar with: "still" meaning "motionless." When Keats repeats the word "still" in line 13, he is actually cleverly bringing both meanings together through the device known as a pun: he wants to hear his girlfriend's breath "still" (a.k.a. "always"), but he also wants to be "still" (a.k.a. "motionless").
But that isn't the only place where the two meanings of the term come into play. Even though the word "still" doesn't appear until line 9, the whole beginning of the poem describes a star which is both "always" in existence and "motionless." (In fact, it's the only motionless star, which makes it special.) So, you could say that, on one level, the whole poem is just one long exploration of these two meanings of "stillness."
How is this Keats's "calling card"? Well, it just so happens that one of Keats's most common themes is the briefness of human life and the desire to transcend that briefness by getting in touch with something lasting, something permanent, something outside of time. One of the most famous poems in which Keats explores this theme is his "Ode on a Grecian Urn" – which you can learn about here on Shmoop. Believe it or not, the first line of that poem calls the Urn "Thou STILL unravish'd bride of quietness." Then, in line 26 of the poem, the speaker talks about "happy love" that is "For ever warm and STILL to be enjoy'd." Coincidence? You be the judge.