The scarf in that pattern is done as a tube -- it's a flat tube, so it's kind of like a normal scarf but with two layers, but it's still a tube rather than two separate scarves.
There are basically two ways of knitting tubes. You can do it flat -- a single-layer long rectangle, really -- and then fold it in half and seam (sew) the edges together. Or, you can do knitting in the round, where you just knit around and around and around to form the tube. (Knitting in the round forms kind of a low-grade spiral.)
If you want to knit in the round, you're going to need circular needles -- I don't know what you have, but I'm assuming you have straight needles (long straight sticks of wood or metal or plastic). Straight needles don't generally work for knitting in the round, because they have a knob or something at the end to keep the stitches from sliding off. (Also, they're straight.)
Circular needles have tips that are several inches long (but nowhere near as long as straight needles), and then a flexible cable wire thing connecting the two. These can be used to knit in the round because both ends are needle-points, so you're basically knitting onto one needle, sliding the stitches across the cable to the other needle, and knitting off of that.
...I'm sure none of that made sense.
There's a video for knitting in the round, though, which will probably make a lot more sense.
In this case, it doesn't really matter which method you do (knitting flat and seaming, or knitting in the round.) It may be easier for you to knit flat, until you're more comfortable with the knitting process.
The main difference for this pattern is that -- you're doing stockinette stitch, which will come out looking kind of like this:
When you knit flat, stockinette stitch involves knitting one row, then purling the next row, then knitting one row, then purling one row, and so on. (Every other row, you've got the Wrong Side [WS] facing you -- this is the side that will be inside once you sew the scarf up. A purl is the back of a knit stitch, and a knit is the back of a purl stitch, so when you're going backwards across the WS row, you need to purl so that the Right Side [RS], which will be the outside, looks like a knit.) If you knit in the round, stockinette stitch involves knitting every stitch (no purling!). So if you find the knit stitch easier than the purl stitch, you might want to look into knitting in the round.
...and I suspect I'm totally not making sense here, so maybe I'll shush for now