Mii (pronounced like the personal pronoun "me") is Nintendo's punning name for an on-screen avatar you can build for Wii games. Generally, people try to build a caricature of themselves, but you can design a Wii to look like anyone you want, from famous people to cartoonish freaks.
A Mii's design is mostly in the face. There are two ways to build a Mii face, you can either refine one of a series of pre-made faces or you can build one from scratch.
Building one from scratch means starting with a pretty generic-looking face and changing the chin, eyes, eyebrows, nose, mouth, facial hair, glasses, hairstyle, and other features to create a unique look. Features can be selected from a menu of options and repositioned and resized on the Mii head.
Refining pre-made faces lets you start from a selection of male or female faces and then selecting slight changes to that prototype, getting incrementally closer to the exact look you want. Once you settle on a refinement, you have the option of going back to the features menu to make any final tweaks you want manually.
Most of the options are in the face. Once you have one you're happy with, you can select a female or male body (skirt or no) and adjust the body's height and weight. The body is much less detailed than the face, consisting of simple shapes with spheres for hands. Finally you are taken to a screen to set you clothing color, birthday, name, and whether or not this will be your primary Mii for playing.
Up to 100 Miis can be stored on the Wii's hard drive for use in various Wii games. Not all Wii games use a Mii. One interesting exception is Super Mario Galaxy, which allows you to use your Mii as your saved game icon. Others such as Wii Sports use your Mii as your in-game avatar, and others don't use it at all. If you'd like to bring your Mii to a friend's house, you can store up to 10 of them on the Wii Remote controller's internal memory to transfer it over.