What are the connotations of "Atzeres Hi"?
Rashi and Ramban #1 (citing the Midrash Rabah): It means to stay back an extra day. It is a parable to a king who invited his sons for a few days of feasting. When the festivities ended, he said to them 'My sons, I ask of you that you stay back another day, because it is hard for me to take leave of you!'
Ramban #2 (According to Kabalah): The first seven days of Succos represent this world, with its various manifestations, and Shemini Atzeres, Olam ha'Ba, 1 where the physical world will come to an end. 2
Rashbam #3: It means to holding back from working.
Seforno: It means to gather together to Daven, to study Torah or to serve Hashem communally in some other way in order to terminate the Chagim with a Simchah of Torah and good deeds. 3
Targum Yonasan: It means to gather together to Daven for rain. 4
Ramban: In similar vein, Shabbos, the seventh ended the creation, and when Yisrael became its partner and kept it, it transcended this world and turned into the number eight (See Ramban).
Ramban: In similar fashion, Pesach, (incorporating the entire seven days of Yom-Tov, represents the creation of this world) and Yom Matan Torah, the eighth day (following the seven weeks of the Omer) - when the physical world would have come to an end, had they not sibsequently worshipped the Golden Calf
What is the definition of "Meleches Avodah"?
Rashi: It means even work that is crucial, that will cause a loss if not performed.
What does the word "Hi" come to preclude?
Rashi: It comes to preclude Chol ha'Mo'ed, on which 'Meleches Avodah' is permitted.