Days between Yom Kippur and Sukkos:
Tachanun: The days between Yom Kippur and Sukkos are days of joy, commemorating the sanctification of the tabernacle. Due to this Tachanun is not recited. From after Yom Kippur until the beginning of the month of Cheshvan Tachanun is omitted.
G-d’s Name: The day after Yom Kippur is referred to as “G-d’s name”. The four days between Yom Kippur and Sukkos correspond to one letter each in the name of Havayah. The first day corresponds to the letter Yud of Sheim Havayah.
Does a Chasan and Kallah fast on their wedding day if it takes place between Yom Kippur and Sukkos?
Yes. However if they feel week they can be lenient and not fast.
Sparks of Chassidus:
The days between Yom Kippur and Sukkos are considered days of growth of the closeness between G-d and His people. Up until Yom Kippur, including Yom Kippur were days of awe, where one does daily evaluation, making oneself a proper receptacle for G-d to have a relationship with him. This is similar to the plowing of a field which makes it ready for seeding and eventual growth. So too Yom Kippur is a day of breaking of the evil spirit through saying confession and recognizing one’s true lowliness in comparison to G-d. This makes one a proper receptacle for the growth which follows. This growth begins after Yom Kippur until Sukkos, and on Sukkos the growth of G-d’s closeness is complete.
 Siddur, as brings Magen Avraham 669/1; brought also in Shaareiy Teshuvah 131/15 and is based on Seder Hayom [in end of section regarding Sukkos], and Kneses Hagedola
The ruling in Shulchan Aruch: In 624/13 Admur rules that no Tachanun is recited between Yom Kippur and Sukkos being that it is days of rejoicing. No mention is made regarding after Sukkos.
Other Opinions: The Shaareiy Teshuvah [ibid] writes that it is not the custom in his provinces to omit Tachanun after Sukkos.
Background of ruling of Admur in Siddur: The Seder Hayom [from the year 1600] brings that Tachanun is omitted until the end of Tishrei because it is a month filled with Holidays and joyous events. It contains four different holidays and we were given a time of forgiveness in this month. He therefore writes it is improper to show any sadness after having experienced all these lofty matters of the month, and rather one is to rejoice in it.
 Sefer Haminhagim p. 126
Some explain this to be because Aseres Yimei Teshuvah correspond to Sheim Havayah while the day after Yom Kippur it does not belong to any name, not Havayah or Adnoy but rather simply “G-d’s Name”. Others [Ashel Avraham Butchach 624] explain it is called this name due to the continuation of saying Hakel Hakadosh. [ibid; See Oatzer Minhagei Chabad p. 256-257 for other reasons mentioned]
 Igros Kodesh Rebbe Rayatz 1/194
 Mateh Efrayim 624/15
 Likkutei Torah Derushei Sukkos p. 79