A Moebius scarf is a scarf with one side and one edge. It isn't hard to knit - the easy way is to knit a strip, flip one end, and sew or graft it together. It's more interesting to knit one as one piece, from its centerline to its edge.
You'll need a skein, or maybe two, of chunky yarn for this (I like Lion Wool-Ease Chunky in 'Wheat'), a circular needle in size 6.5 mm (US 10.5), and a double-pointed needle of about the same size.
Make a slip knot, put it on the circular needle, and cast on 100 stitches using the backward loop (half-hitch) method. When you have those on the needle, make sure they aren't twisted, put a marker on the needle, and then start picking up the loops on the other side, between the cast-on stitches. (Yes, this puts a twist in – but you need that half-twist for this to be a Moebius.) Pick up a hundred more stitches this way, and put another marker in for the end of your round.
You'll need to use the double-pointed needle for the first two or three rounds, because it's going to be tight on the circular needle, and you also need to take out the twist in the cast-on loops.
Work in garter stitch, seed stitch, moss stitch, or any other reversible stitch that will fit evenly in 200 stitches, until you have something that's six or seven inches across the space between the loops of the needle, then bind off. You'll probably have to interchange knit and purl stitches at the middle of the round to make the stitch pattern work.
To wear this, put it on with the knitting flat against your neck, which will put the twist over your chest. Flatten the twist into a fold, and put your coat on over it. The Moebius won't fall off, and your neck will stay warm.
The Yarn Harlot's stitch pattern for Encompass works very well on a Moebius.