One of the the best things about Skyrim (or any TES game for that matter) is that you don't have to become one of the three archetypes, or stereotypes in the cases of most hastily put together RPGs. My personal favourite is the Archer, and I'll state why as well as give my personal thoughts on the other three:
Warrior: Melee is superior in most cases of the Elder Scrolls games because of the huge amount of weapons, how many ways to go about (Tanking, assassin, swashbuckler, barbaric) and it's also the most developed when it comes to talent trees. Melee users get the most powerful weapons from Daedric Princes, and benefit mostly from magic.
Archer: I love shooting in this game more than the previous games because of the precision of bows as well as the sound they make (for once it's realistic). Archers can also move about a bit, which to me is one of the basic requirements for combat - having to 'strategically' stand to cast spells and shoot is not my idea of fun. You're far away, the noise is minimal and you can slow enemies who find out your position. Shame you can't shoot enemies in the legs and cripple them, nor the arms and disarm them and such.
Mage: The worst archetype in the game for how limited magic is. There could be a million spells and only fire would be worth using. Magic in this game is a joke because there are three effects - fire, which increases damage dealt (and burns enemies); shock spells which reduce an enemy's mana (useless against non-player characters) and healing. That's as far as my knowledge for magic goes because it's useless. Dual-casting is interesting and being able to use magic and melee is a definite plus from me for this game.
After looking at this question I might go and play some Skyrim with a new character!