
There it is! Done!...
No. Ok, I usually start out a mob icon or sprite by deciding what perspective I'll be using. Normally, it's pretty important when first designing a sprite to think about where it will eventually end up; In other words, what will the background tiles be and what other sprites will it be shown with. In the case of choosing a perspective for this sprite (since we're working on the pretense that this sprite will be in a game) I want it to have the same perspective as all the other sprites and background tiles. However, since this will be the first sprite we've got a clean slate and, more importantly, we'll be choosing the perspective of all the other sprites for the game here.*
*In my opinion, it can't be stressed enough: keep a consistant style across all artwork in a game! It's visually very jarring to see two sprites with different perspectives alongside one another. As that applies to perspective, it goes for every other aesthetic and informational aspect of your design: colors, lightsource, scale, and drawing style (cartoony, photorealistic, iconic).
For perspective, we've got a few standard choices:
Front or Side View:
,
Three Quarters: (I believe it's called)
,
and Top Down or Bird's Eye:
Front view is one of the easiest to draw mainly, I think, because that's the perspective from our eyes (most of the time) and the way most learn to draw. It's really good for side-scrollers and platformers. Top down is pretty rare, but does occur in most shoot-em-ups (like 1942) and others (like Gauntlet). I chose 3/4s, primarily because, when put in front of 3/4 perspective backgrounds it gives the player a good 'god' view: above (about 22.5 degrees above) and slightly in-front of what's going on. It also makes it fairly easy to tweak it into the more formal isometric style which I've been concentrating on.
There are other perspectives: orographic (including isometric - a more specific and regimented form of 3/4s), 2 and 3 point perspective (distant objects appear smaller - very hard to do in 2d pixel work), and more.
So, the short version is: I placed the squashed X to kind of give a refrence for what perspective I was going for and made the center of it right around where the zombie feet will go.

Here, I'm sketching out the basic form. I usually use just one color for this (any will do). This is where some fundamental drawing and sketching skills can really help you. Anatomy (the relative size of different parts of the figure) and dynamics (the pose and 'action' of your figure) are things to keep in mind. If it's humanoid, consult some online or hardcopy sources on human anatomy especially if your doing a sprite that is anywhere near a realistic stlye. The dynamics fo your pose is up to you, but try to make it obvious what limbs are what and representative of what the character might do.
That being said, with the eraser and undo button not far away, don't put too much detail and time into this part. You're trying to establish the 'essence' of the finished drawing. You're not doing line art. Sketch. Go quick and flow.
Note: sketching an outline isn't the only way you can set up this step. You could use a sillouette, a wireframe (good for later animating), or any other method that sets a framework for later detailing.

Now we've got a pose we like and anatomy seems ok. We'll alternately select the background color and erase pixels we don't want on the figure, and select the sketch color again to redraw some of what we've erased. I use this step to refine the sizes and volumes of arms, legs, torso, etc. Here, I've dried his bowels up and given him the waist and abdomen of the not-so-newly dead. The rib cage becomes more prominent. Normally I'd remove that stray pixel on the (our) right side of the head, but I like the way it makes the head seem ragged and I want this corpse torn-up, so I leave it. The hands and forearms are a lil' big but I like the way it makes his arms more paw-like, so I thin up the upper arms even more to increase that contrast. I thin the legs, too (no rich Mensa cerebellum buffet around this guy) and move the (our) right foot back a bit to offset his stance.
You might also notice I've used the sketch color to fill in the shoulder on our left, the ribcage and under the arm. It would seem I've decided on a light source and these are shadows (of the head, chest, and arm respectively). In this case, the light would be above (almost 80 degrees above?), slightly in front of him, and to our right. This just came from me sketching and wasn't really planned, but that is my most common placement for my light sources. Some people would call this a rut. I call it a groove.

Here I ditched the X and kept cleaning up and refining the sketch. I added a shadow on the (our) left arm as well as an 'under' shadow on the other arm. I also added shadows on the forward shin helping to indicate that leg is bent slightly. I liked the fluttering piece of head skin so much I lengthened it and added one on the opposite shoulder. I also started cleaning some of the lines (making them only one pixel wide) around the other shoulder.
I should mention here that's it's a really good idea to switch back to viewing your piece actual size frequently if you're working in a zoomed view (and you should be, damnit!). Zoomed pictures can often look verrrrry different than the actual size: details you thought looked good look like someone smeared bacon on the monitor, way too many colors, etc. If you use Graphics Gale (and I recommend it), you're lucky because there's a preview window that shows your sprite actual size as it's being drawn. Otherwise, switch back and forth as often as you comfortably can.

So here, I choose my colors. Now, the challenge was for 16 colors, but there's absolutely no reason to choose all 16 here. In fact, it's a bad idea to do so. Now even tho BYOND sprites and tiles can use 256 colors, if you're ever resticted by some other reason (like the challenge rules), hold some of your color choices in reserve to use for blending between colors, highlights, special stuff, etc. I chose a very unsaturated cyan (a few shades of it) and slowly 'hue-shifted' (changed colors as well as brightness/shade over a couple of pallette colors) one of those blues to a grey-brown (all on the lower right; my sketch color is in the very far lower right).
I then took my darker color and started to dot in some heavy shadows
By the way, don't get too intimidated with color choice. There's a couple of different ways to go back and change the colors depending on your program, so don't feel you'll be too commited. I actually find this step to be one of the funner parts of it all.

Yeah, I got a little bored here and started to add facial features. Normally, I'd suggest using more than one pixel for eyes (they should be treated as depressions in the face as far as shading goes especially on the shadow side), but I thought the extra small eye effect was good and creepy here. You'll maybe notice I didn't use the cyan (which I had reasoned was close to dead-flesh color) and instead used the tans/browns. Like I said above, don't get married to your colors. Play around with them.

Here's an example of going down a road and realizing you ended up somewhere you didn't want to be. I thought it might be cool to give the zombie a suit (as in that what he was buried in), so I gave him a businessman's standard grey 2 piece suit with power red neck tie.
Then I decided this looked like complete and utter crap. What to do? I could 'undo' all the way back to step 6 or I could reload one of the saved versions. Say what now? What saved versions? Wellllll...It would take me getting to step seven to tell you to: save your work often and with different names. For instance, each one of these steps is saved as "zombeh-01.png","zombeh-02.png", etc. This allows me to move back and undo whole steps or draw out crazy ideas and not worry about ruining anything. This is especially true if you prefer using MSPaint, which only has three 'undo' steps back.
But, I played around with it a little more, tearing the suit up and removing a pants leg torn off by a hungry dog. I even added some blood on the hands, shirt, and floor. And, since I was free to play around because I saved the steps, I messed around with the face and head, adding a gaping lower jaw and torn eye socket.

I reloaded step 6 and decided I'd go with a classic kindof 'Creepy Magazine' sorta look by having just some torn pants. I dunno, at least this keeps it kind of time-generic and the icon could be used in present day or some medieval fantasy setting. I chose a dark brown for this 'cuz I thought it looked good with the blue. I also continued to 'flesh' out (HAHA.ME FUNNY) the chest.

At this point, I'm ditching the sketch blue and using shades of the brown on most of the outlines for the skin. The very light tan used as a main color is outlined by a darker shade even where there would be no shade (like his left shoulder (our right)). What gives? There wouldn't be shadow there, you say. Well, shaddup. Sorry. It's actually an outlining technique called 'selout' or selective outlining. I sometimes do completely border a sprite with a fully black outline (and that works sometimes), but this technique allows me to outline the sprite and still provide another level of shading. The shoulder is a lighter outline than I'm using on the lower forward leg and this difference in shade allows me to express the lightsource hitting the figure while still retaining an outline that defines form.
A very simplified form of selout on a circle might be:

Also, in this step I shaded the chest and right shoulder a little differently. I'm using the light blue (2nd from the right, bottom row) instead of the light tan next to it. I'm doing that to add light from a second (bluish) light source coming from his right side. Those two colors (light blue and light tan) aren't really that different in shade, are they? There about the same brightness, but they're different hues (the H in HSB and the left right difference in a color picker). The blue light will hit the figure on his right side (and all the shapes and sides of individual sections of his right side) and be tinted a little blue.

Shaded the pants a bit here and the shadowed palms of the hands. I also added some more blue in and around the outlines to be blended in with the tans later.
I'll admit, I wasn't too thourogh with anti-aliasing and consistent borders on this piece, and as much as anything, it's because I wanted the zomb to have a lot of texture and 'feel' ragged. If I had been more consistent with some of the color blending, he would have looked smooth and rounded. That's my story anyway, and I'm sticking to it.
Along with that I added some spare pixels to his right head to match the torn bit in his pants. The two horizontal pixels gave kind of a wind blown look to the hanging skin and cloth, I thought.

Pretty satisfied, but I was thinking that he wasn't nearly gorey enough for a zombie here, so I added some blood, using ideas I stole from the aborted pic in step seven. Two shades. Added some stray red pixels to indicate blood spattering.

When you're working with a solid background (as I often do), in this case white, it's a really good idea to change it up once and a while to a different color (or, if you've got some of the game's background tiles, use those). This will give you an indication of how your outline will fit against various backgrounds and whether it needs to be adjusted.
I decided against the wind thing and removed those. For fun, I added a pool of blood at his feet (which I also fixed a bit and made flatter).

Noticed in 12 that the palms had lost their depth, so I added a darker shade inside them. Cleaned up the outline. I tried something with shading the eyes here I really didn't like, so that goes away in the next step. Also here, I was playing around with a possible (very non-game tile) background for the challenge. This addition isn't really helpful for our purposes, but it did turn out to focus in on two things that may be useful to know:
My god, the zombie farthest to our right must have had elephantitis of the head!

I took that oogly background out and decided it worked best as a transparent sprite. Here, I switched my brain back to game mode and took out the pool of blood. It would have looked pretty damn bad if that puddle was following the moving sprite around always in the same spot.
I thought It looked pretty ok at this point so I saved it and started writing this. :)