Everyone's Minecraft character starts off looking the same. However all you need is a simple graphics program and you can personalise their appearance. Login in at minecraft.net and select the Preferences link. Download the char.png and start editting. The image at Skin (article) - Minepedia - The Minecraft Wiki!
shows you what each region of the image corresponds to. When done, use the Preferences page to upload your skin onto Minecraft. You should soon see the image on the screen changing to your new appearance. This will be what you look like every time you log into a Minecraft game. Note that you can also view other people's skins if you know their username. Please respect their artwork.
Yes, it is very blocky but that is the nature of the game. The one optional addition to the skin is the "hat" (or head accessory). This is the area of the skin which is normally blank, but if you start to fill it in, you can give your character a hat, a mask, glasses or even make his head a little larger.