Learn the history behind this Hebrew blessing/greeting:
The Ashkenazi common greeting on Rosh Hashanah is שָׁנָה טוֹבָה “Shanah Tovah“, which, in Hebrew, means “[have a] good year” or similar greetings. Thus, in Yiddish the greeting is אַ גוט יאָר “a gut yor” (“a good year”) or אַ גוט געבענטשט יאָר “a gut gebentsht yor” (“a good blessed year”). Sephardic Jews traditionally say “tizku l’shanim rabot” or “[anyada buena, para] munchos anyos“, in Ladino, both of which mean “[have a good year for] many years”.
Serious greetings and blessings, based on the nature of the day, commonly used among religiously observant Jews are כְּתִיבָה וַחֲתִימָה טוֹבָה “Ketivah VaChatimah Tovah” which means “[may you be] written/inscribed and sealed [for a good new year i.e. by God].” After Rosh Hashanah ends, the greeting is abbreviated to גְּמַר חֲתִימָה טוֹבָה “G’mar Chatimah Tovah” (“[may you be] finally sealed [for a] good [year by God]”) until Yom Kippur. After Yom Kippur is over, until Hoshana Rabbah, as Sukkot ends, the greeting is גְּמָר טוֹב “Gmar Tov” (“[a] good conclusion [of God’s judgment]”).
The above describes three important stages as the spiritual order of the Ten Days of Repentance (the ten days between Rosh Hashanah and Yom Kippur) unfolds: On Rosh Hashanah God “‘opens’ the ‘books’ of judgment” of creation and all mankind starting from each individual person, and in those books it is first “written” what will be decreed, hence the emphasis on the “ketivah” (“writing”). The “judgement” is then “pending” and prayers and repentance are required. Then on Yom Kippur, the judgment is “sealed” or confirmed (i.e. by the Heavenly Court), hence the emphasis is on the word “chatimah” (“sealed”). But the Heavenly verdict is still not final because there is still an additional chance and positive expectation that until Sukkot concludes there is hope that God will deliver a final good and favorable judgment, hence the use of “gmar” (“end”) that is “tov” (“good”).