The round spiral is known by a few names, apparently. I was surprised to read it referred to as a "turban challah," which to me brings up all sorts of funny visions of Maimonides (Rambam, Moshe ben Maimon, however you want to to call him).
According to the Encyclopedia Judaica, the New Year's spiral is a shape with a Ukrainian origin, originally a bird shape with the center of the spiral culminating in a bird's head: "The bird's head symbolizes the phrase in Isaiah 31:5 'As birds hovering, so will the Lord of Hosts protect Jerusalem'" — which helps to explain why this spiral shape would be called a faigele, "little bird" in Yiddish.Regardless of what you call it, it's lovely. The following technique is a great way to make a really nice, smooth round challah shape. It works equally for plain challah and for stuffed challah, so I've given you a set of pictures from both methods.
Start out with your dough on a flour surface. With a rolling pin, roll the dough out smoothly and flat into a long, oblong rectangle about 1/4 inch thick. Next, coil up the challah, trying to keep as many of the bubbles out as possible. The resulting strand should be about the size you would normally use to braid challah.