On most days when Avinu Malkeinu is recited, it is included during Shacharit and Mincha on that day. But on Yom Kippur, Avinu Malkeinu is also recited during Maariv and Ne'ila, except when Yom Kippur falls on Shabbat, in which case, Avinu Malkeinu is recited during Ne'ila only. Other than Ne'ila, Avinu Malkeinu is never recited on Shabbat.
Throughout the Ten Days of Repentance, five lines of Avinu Malkeinu that refer to various books include the word Katveinu translating to "inscribe us." During Ne'ila, this is replaced with Chatmeinu, translating to "seal us."