Skins are a collection of 2D textures that are applied to the base avatar's prims in very specific ways. There's six templates for avatar skinning. Head, chest and arms, hips and legs, eyes, hair, and skirt. The official design and textures forum is where you'll find conversation about designing skins and clothes. Here's a nice overview of clothing template basics.
You can get the official Linden templates here. There are other popular free templates; check out the ones by Chip Midnight and Robin Sojourner.
Most people seem to use Photoshop or the freely available GIMP to do their texture editing.
Transparent textures don't always act as you might expect. Use one on the base avatar's skin and the default skin will show. Use one on basic avatar hair to create a bald avatar. Use one on basic avatar clothing and you'll see the layers of clothing or skin underneath.
To make skin or clothing with your name as creator, you must start with a new one in your inventory. You can right-click in inventory and create new clothing or body parts, or you can do it from the Appearance Editor.
Seeing odd colored edges on some of your textures? Try blurring the colors over the outlines - color outside the lines. Sometimes you'll get an odd line or plus sign on the back of the head, try pulling the mouth texture further inside the outlines.
To apply a texture to the avatar's skin, go into Appearance mode, choose Skin, then in the bottom left you'll see "Apply Tattoo" boxes for the head, upper body, and lower body. Place the appropriate textures into these boxes and save your new skin.
To apply a texture to a piece of clothing, go into Appearance mode, choose Shirt (or other piece,) choose New, and apply your texture. Then manipulate the clothing sliders. Save.
Tattoos are a design on an otherwise transparent texture. They're often sold as undershirts or underpants so that they can be applied to an avatar already wearing a skin texture.
There's only one arm and one foot on the templates. Both arms are textured to appear identical, the same is true of the feet.
Clothing and skin textures are shrunk to 512x512 pixels when displayed. Avoid uploading larger textures for these uses - you and your graphics editor will do a better job shrinking the texture than SL will.
You'll need to know how to use layers in Photoshop, the GIMP, or whatever you use. If you don't know, stop now and find a tutorial on the internet. Grab both Robin Sojourner's and Christiano Midnight's templates off the internet. Open them in your graphics tool, copy the layers and add them to what will become your own templates.
Go to the directory where your Secondlife.exe is installed. Go into the "character" directory. There are several *.tga files here. Most of them are alpha layers used for making the default skins. Copy these into your templates as layers.
Natalia covers the basics of working in layers with PSP and GIMP here. I don't know how to make a layer into an alpha layer in PSP. In GIMP, you use Layer | Transparency | Color To Alpha and pick the color that'll become your "invisible" - it'll either be 000000 (black), 888888 (50/50 gray), or ffffff (white) depending on which layer you're using. You can adjust the transparent-ness of the layer by adding a mask and changing the black/white color of the mask.
Over time, you'll find you want to make adjustments to the layers. Maybe certain ones will become just for male or female skins, or you'll make your own custom layers. Eyelashes aren't shown on some templates. They're determined by what's in the top right corner of the head texture. If the corner is transparent, the default black lashes are shown. Some of the layers are meant to leave a transparent place in your skin. I apply the eyelash alpha directly to my base skin layer - this works well most of the time.
There's several photoshop tutorials on the web about skin. I like this one.
Eloh Eliot has released several skins as templates for you to experiment with. Here and here. I put up my freebie gray neko skin here.
The test grid is a wonderful place to test out skins. Nothing you upload or create there comes with you to the main grid, but everything you upload is free since your real linden balance is not affected by what you do there. Currently, you add --aditi after Secondlife.exe on your command line to get there.
To use a texture, go into Appearance mode, choose eyes, place your texture into the "applied texture" box, and save.
When creating eye textures, you want to rotate them 90 degrees counterclockwise - SL automatically rotates textures 90 degrees clockwise on eyes. Eye textures shouldn't be larger than 256x256 pixels.
Here's some tutorials for making eye textures. Kim's, Kris Costa's. The texture is applied to both eyes. To make different colored eyes, attach an appropriately textured prim over one eye. You can use prims over eyes to create "glowing" eyes with fullbright.