Ok, I might be wrong about this because I dont have the game but I want to start a mage. I saw the armor on a youtube vid and I was wondering if they could wear heavy armor? Can mages wear heavy armor and not the cloth stuff?

Yes. Armor doesn't restrict magic abillities, so you can wear whatever you want.

of Course they can my character is wearing legendary Ebony armor full suit and He's a Arch-Mage/Archery/Dual wield Swords/Champion and i use Magic with him all the time, my heavey armour is about 50 and light is 79.
you just need to skill some points in to the armor perk and you'll be fine.

there are no 'classes' in Skyrim. You do not select or make a class archetype

you dont even need to put perk point in Hvy armor skill. Don't want to carry all that weight, find and use the Steed stone. Not only does armor you are weght nothing, but it adds 100 lbs to your carry limit.

Isn't the Armor perks designed to increase the damage you can take, there for
you would want to level up in it.

Depends if your point is really defense, I get a sense this thread maybe more about style in which case it wouldn't actually really thus matter. Mage's don't stand on the front line and the steed stone removes the encumbrance of it, personally not the way I'd go. I would go for the perk as the Atronach or Lord stones are much better since mage's usually have low HP and thus usually very vulnerable to magic themselves. Currently tho I'm use the mage's stone for faster skill-ups on my mage... I'll switch at some point but not sure when.