What are the connotations of "Atzeres Hi"?
Rashi (from Midrash Rabah): It means to stay back an extra day. A parable for this is 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 Sukos 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: It means holding back from working.
Seforno: It means to gather together to pray, 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 pray 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 subsequently worshipped the Golden Calf - and like the intermediate days of Sukos, the period in between Pesach and Shavu'os are considered Chol ha'Mo'ed. And this explains why Chazal refer to Shau'os as 'Atzeres' (See Ramban). Refer also to 23:36:1:4*.
Seforno: As the Pasuk indicates in Tehilim, 149b. See also Iyov, 1:5. See also Seforno as to why the Torah (in Devarim 16:8) also refers to the seventh day of Pesach as "Atzeres", and why the Chachamim refer to Shavu'os as 'Atzeres' and why the Torah does not. Refer also to 23:36:1:2**.
Seeing as the rain season is due to begin in two weeks' time, on the seventh of Mar-Cheshvan
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.
Rashi gives a parable - Hashem tells Yisrael to stay another day, for it is hard for Him that we leave. Why is this only at the end of Sukos?
Hadar Zekenim: In the parable, the king asked when they will return. When he heard that [after Sukos] they will not return until seven 1 months, he asked them to stay another day.
Perhaps this is when they know that it will be a leap year. Even in a regular year, when it is only six months, we stay an extra day! Perhaps the text should say 'six', or 'the seventh month'. (PF)
Megilah 3:5 calls Shavu'os 'Atzeres'. Where do we find that it is called Atzeres?
Hadar Zekenim #1: The Targum of "bi'Shavu'oseichem" (Bamidbar 28:26) is "b'Atzartechon."
Hadar Zekenim #2: Because Yisrael received the Torah on Shavu'os, they were Ne'etzaru (held back) from their wives [beforehand], therefore it is called Atzeres.