According to halachah, a succah must be made of at least two complete walls and one smaller wall, the length of a tefach. The holy sefarim explain that this represents an arm, which is made up of two larger parts and a tefach, the palm. These walls of the succah signify that when we sit in the succah, keviyachol, it is like Hashem is hugging us.
It states (Shir HaShirim 1:4) חדריו המלך הביאני, "The king brought me into His [innermost] chambers. בך ונשמחה נגילה, "We will rejoice with You." There is joy in the succah. When we are in the succah, we are in Hashem's innermost chamber, a place of immense joy. As Chazal (Chagigah 5:) say, after the churban Beis HaMikdash, Hashem cries in His outer chambers and is happy in His innermost chambers.
There are several comparisons between the succah and the Beis HaMikdash. The Pri Megadim states: One mustn't use a succah as a shortcut and it is certainly forbidden to spit there because the succah is holy and has laws similar to those of the Beis HaMikdash.
Arizal teaches that one's table in the succah should have four legs to resemble the Shulchan of the Beis HaMikdash, which had four legs.
Chazal consider the possibility that a succah needs to be built on Har HaBayis, next to the Beis HaMikdash. This theory is disproved in the end. Nevertheless, a אמינא הוא, an initial thought of Chazal, also has significance, and it points to the holiness of the succah. Its holiness is similar to that of the Beis HaMikdash, which is built on the Har HaBayis.
Some people don't sing (or say) "Shalom Aleichem" on Friday night of Succos. Shem MiShmuel explains that the kedushah of the succah is so great that malachim cannot enter there.
