There are two kinds of single round stripes you can knit: helix and bull's-eye. Both ways of knitting single round stripes are equally good. The main thing is that you do not want to combine both types of stripes in the same piece of knitting. Doing so will result in wonky stripes. Therefore, it's important to know how to knit the stripes both ways.
For helix stripes, the two colors spiral around each other and there is no carry of the yarns up the back of the work. In large diameters, helix stripes look just the same as bull's eye stripes. The spiral effect becomes more pronounced at smaller diameters. If a series of decreases are worked, as in the crown of a hat, a spiral design will appear.
At the beginning of the round, make sure the old color is going off to the right when you drop it. Then pick up the new color and knit the first stitch. Do NOT cross the old yarn with the new yarn. The old yarn should not be trapped in the first stitch. Continue knitting to the end of the round. Repeat the process for each additional round.
For bull's eye stripes, the two colors are stacked on top of each others like a stack of rings. There is no spiraling at smaller diameters. If a series of decreases are worked, as in the crown of a hat, a bull's eye design will appear. The yarns are carried up the back of the work.
At the beginning of the round, take the old color and carry it across the back from the last stitch to the first stitch. Then pick up the new color behind the old color and knit the first stitch. Check that the old yarn is trapped in the first stitch of the new color and then drop the old color. Continue knitting to the end of the round. Repeat the process for each additional round. When picking up the the new yarn, be sure to keep the carry up the back loose.