A frequent instruction in knitting, and one which is especially confusing to newer knitters, is "work the stitches as they appear." Sometimes this direction is phrased in a different manner, which also means the same thing: "knit the knits and purl the purls." What, exactly, does this mean?
What these instructions are asking you to do is to work each stitch as the type of stitch it appears to be as you come to it. In order to understand that, and interpret it correctly, you need to have a couple of tools.
First, you need to understand that a knit stitch is the reverse of a purl stitch. This means that if you knit a stitch with one side of the fabric facing you, you get exactly the same result as if you had purled the stitch with the other side of the fabric facing you, and vice versa. When executing the "as they appear" instruction, it only matters what the stitch appears to be from the side you are currently working on, and not what it was originally worked as.
Second, you need to know how to recognize a purl stitch and a knit stitch when you see them. Each stitch which is currently on your needle will look like an arch sitting over the needle, regardless of which type it is; what really tells you which type is how that stitch emerges from the stitch below. Purl stitches are worked by pulling a loop from in front of the fabric through to the back, and therefore they emerge from the back of the stitch below, which leaves the top of that arch sitting in front of the fabric, giving you the characteristic "purl bump." Look for this bump up against the bottom of the needle on your purl stitches. Knit stitches, on the other hand, are worked by pulling a loop from behind the fabric and out the front, emerging from the front of the stitch below, which leaves the bump on the back of the fabric, and from the front looks as though the legs of the current stitch just stick smoothly down into the stitch below.
Therefore, when you are knitting the stitches as they appear, you are going to (1) recognize the type of stitch you are seeing, and (2) work it as the same type of stitch. Ignore what you know about how the stitch was originally worked -- it isn't relevant. For instance, if you're doing ribbing (one of the common places where this type of instruction is used), and you end a row with a purl stitch, when you turn the work around, that stitch will look like a knit from the side you are now on. That is what matters, not that it was originally a purl; work that stitch as a knit.
As I just alluded to, this type of instruction commonly occurs in ribbing, which is formed by working aligned columns of knits and purls. It's also quite common in cabled patterns, and sometimes in simple laces.
Here's a little exercise to make sure you've got this concept down. Get yourself some nice smooth yarn (this is not the place for fun fur! you want to see what you're doing) of your choice, needles that are appropriately sized for the yarn (if you have a choice between slightly big and slightly small needles, go for the small one -- this will be easier to see in a slightly tight swatch), and a 6-sided die. Cast on about 30 or 40 stitches. Now, for your first row, roll the die once, and knit that number of stitches. Roll it again, and purl that many. Repeat this across the row, rolling for the number to knit and then again for the number to purl, until you run out of stitches. You can put the die away now; you're done with it. Turn, and work each of the stitches as they appear. Continue to work in this fashion until you've got a couple of inches in this swatch, and then take a good look at it. If you've done it correctly, you should have aligned columns of knits and purls, giving you clearly defined ribbing with ribs of random widths.
