Shape of the Challahs:
It is customary to form the Challahs for Shabbos as a long straight dough. This is opposed to round or square. The reason for this is because the Challahs represent the letter Vav of the Tetragrammaton. Alternatively the two Challahs in the shape of a Vav represent the twelve showbreads of the Temple, as Vav is Gemtria of six.
Rosh Hashanah: The ancient custom of Ashkenazi Jewry is to bake round Challahs in honor of Rosh Hashanah. This symbolizing the roundness of a crown and corresponds to the coronation of Hashem on Rosh Hashana. Alternatively it is done as a symbol of good omen as all round items due to not have a start or finish and hence reflects longevity of life. Likewise round represents unity. Some write that this applies only on Rosh Hashanah itself, however on Shabbos Shuva one is to return to the accustomed shape of the two Vavs.
Braid: It is customary to braid the Challahs baked for Shabbos as in the past it was common to add meat gravy to the dough and hence the Challah required a sign that would remind one not to eat it with dairy. This custom has remained today as well despite that the Challahs are left Pareve.
 Elya Raba 167:2 in name of Shlah. The five fingers on each hand which holds the Challahs represent the two Heis of the Tetragramottan, and the Challahs form a Yud when held. Hnece the Vav of the shape of the Challah complete the Tetragramottan. [Shem Havayah]. [ibid]
 Likutei Mahrich in name of Divrei Chaim
 Piskeiy Teshuvos 583 footnote 46; 242 footnote 105; Otzar Minhagei Chabad 129
 Brought in Piskeiy Teshuvos 583 footnote 46
 P”M 242 M”Z 1; See Yoreh Deah 97:1; “A Semicha Aid for the Laws of Basar Bechalav” Chapter 97