The name comes from the book of Daniel. In his night vision, the prophet sees thrones set up, "and an Ancient of Days sat; His garment was white as snow, and the hair of His head like pure wool; His throne, flames of fire." Atik Yomin, in Aramaic, means "the Ancient of Days." Massekhet Atzilut already applies this verse to Keter, the first Sefirah.
In Etz Chaim, Atik is the first and most hidden of the partzufim of Atzilut. Gate 3 describes it as a link: the Malkhut of Adam Kadmon descends and clothes itself in Atzilut to bind Adam Kadmon to this world, and "this aspect is called Atik Yomin, for these are seven days, ancient, coming from the Malkhut of Adam Kadmon." Atik then becomes the soul of Arich Anpin, which is the Keter of Atzilut.
Why "of days"? Etz Chaim explains: of Atik, only its seven lower Sefirot are revealed, which are like the seven days of the beginning, and they appear in the seven arrangements of the skull of Arich Anpin. "This is the secret of what the Zohar calls Atik Yomin: it is revealed only through the aspect of the days." And it is insofar as it clothes itself in Arich Anpin that it is written of it "the hair of His head like pure wool," although this is not really its head. Its first three Sefirot remain above, a surrounding light that Arich Anpin cannot contain.
Shaar HaKlalim distinguishes this way: the most hidden head is called simply Atik, and the seven other Sefirot, from Chesed to Malkhut, are called Atik Yomin. Etz Chaim also connects to this level the "head that is not known" of the Idra Zuta, of which one does not know what is within it. Chaim Vital warns: in this head there is no grasp at all, and one cannot speak of it at length.
Cordovero, for his part, does not separate Atik and Arich Anpin. In the Pardes, Atika Kadisha, "the holy Ancient One," is Keter, Arich Anpin: "the word Atika indicates precedence, as in the verse of Daniel," and "holy" means separate, raised above Zeir Anpin.
Keep the image that both schools draw from Daniel: whiteness. Etz Chaim specifies that these white hairs, although they are the root of judgments, are "more mercy than the mercy of Zeir Anpin." At the very top, even severity is white.