Agility controls your chance to hit enemies, your chance to dodge their attacks, and your maximum Fatigue. Also reduces your chance to get knocked down in combat.
Governing Attribute for Block, Light Armor, Marksman, and Sneak. It affects:
Your ability to successfully hit enemies with a weapon and dodge their attacks (not magic).
Your ability to use the Sneak (including pickpocketing), Security, Acrobatics and Block skills.
Your maximum Fatigue.
Your resistance against staggering.
Endurance controls your starting and maximum Health, and your maximum Fatigue. Also slows down fatigue loss while running, and while fighting.
Governing Attribute for Heavy Armor, Medium Armor, and Spear. It affects:
How much maximum Health you have.
How much maximum Fatigue you have.
Intelligence controls your maximum Magicka, and the amount of magicka gained per hour of sleep.
Governing Attribute for Alchemy, Conjuration, Enchant, and Security. It determines your base amount of Magicka.
Both Enchanting and Alchemy are enhanced by higher Intelligence.
Personality controls how much people like you, and the prices you get at vendors.
Governing Attribute for Illusion, Mercantile, and Speechcraft. It affects:
Your base disposition with NPCs.
Your ability to persuade people by any means.
The ability to bargain, in tandem with the Mercantile skill.
Speed controls your rate of movement when walking, running, swimming, or levitating.
Governing Attribute for Athletics, Hand-to-hand, Short Blade and Unarmored. It affects:
How fast you move in the game world, which includes swimming.
Strength controls the damage you cause with weapons, as well as how much you can carry. It also helps determine your maximum Fatigue and starting Health. Strength also factors into how your weapon durability degrades on each successful hit: Higher strength means higher weapon degradation.
Governing Attribute for Acrobatics, Armorer, Axe, Blunt Weapon, and Long Blade. It affects:
How much damage you do with weapons, including Marksman weapons, and how much damage you do to your own weapons when you perform a successful attack as a result.
How much you can carry. This, in turn, affects your speed and jump height.
Your starting Health
Willpower controls your odds of success at spellcasting and your chance to resist Magicka. It also determines your maximum Fatigue.
Governing Attribute for Alteration, Destruction, Mysticism, and Restoration. It affects:
Your ability to successfully cast spells of any College. This includes Conjuration and Illusion, even though they are 'governed' by Intelligence and Personality respectively.
Your ability to innately resist Paralyze and Silence magic.
Your maximum Fatigue.