Games have always been about the action, the gameplay, but as technology advances, video games have pushed computers and consoles with ever more complex graphics. As characters on the screen moved from pixelated figures to fully rendered 3D models, they've become more lifelike, and along with better character appearances, game developers started adding more ways to customize the character you would play, more so than just in stats, but in looks as well.
Elder Scrolls IV: Oblivion
Character customization options started to appear in role-playing games first, like the Elder Scrolls series, and also in MMO games like EverQuest and eventually World of Warcraft. In the early days, 3D models were low polygon counts and looked quite crude. You had a choice between several different faces, and then options for skin color, eye color, and hair.
World of Warcraft
In the case of the popular MMO introduced in 2004, characters still look quite blocky, but Blizzard had the good sense to use very stylized art, instead of going for a realistic look. This makes the resulting characters look iconic, and not too dated. Again, you mostly had a choice of a couple of faces and hair styles, along with piercing options.
A big advance was made when we moved from a series of pre-made models, to sliders, allowing the player to apply morphs to their characters, and get a much larger selection of appearances. Mass Effect was a popular example of that in 2007.
APB introduced in 2009 a very innovative character customization system, that allowed players to change almost everything about their characters, using sliders instead of pre-made faces. This is one of the most in-depth customization system made so far, allowing you to adjust any small detail, including hair length, muscularity, veins, freckles, and so on. Tattoos can even be designed and added in the game.
Saints Row the Third
Saints Row has always had a very advanced character customization, and just recently released their preview for what character creation will look like in game. Again, they use sliders to go through the various options, although there isn't as many options as in APB.
Mafia War 2
It may seem weird to add a Facebook game after a list of RPG and MMO games, but the fact is character customization options are no longer restricted to role-playing, or even high budget games. Now often action games offer some basic appearance options like the hats and special weapons in Team Fortress 2, and even social games, which traditionally have only offered you a basic gender selection, now feature full customization options, even on sprite graphics. Mafia War 2 is a good example, offering different skin types, tattoos, hair styles, and a large collection of clothing options.