What are the properties of each stat point?

Strength

• Each point increases Barbarian's damage by 1%
• Each point increases armor by 1 for every class

Vitality

• Each point increases your Life (dependent on your level, higher level=more vitality per point. See below for formulas) for every class

Life when player level < 35 = 36 + 4 × Level + 10 × Vitality

Life when player level ≥ 35 = (level - 25) × Vitality

Source

Intelligence

• Each point increases Wizard's and Witch Doctor's damage by 1%
• Each point gives .1 to all resists for every class

Dexterity

• Each Point also increases Monk's Armor by 1 with the Seize The Initiative passive unlocked at level 20.
• Each point increases Monk's and Demon Hunter's Damage by 1%

• Each point increases chance to dodge for every class

    Dex range     Dodge% per point of Dex
  
   1    - 100             0.100
  
   101  - 500             0.025
  
   501  - 1000            0.020
  
   1001 - 8000            0.010
  

Source

From Diablo 3's Game Guide:

• Strength: The primary attribute for Barbarians. Strength increases Barbarian damage, and increases Armor for all classes.
• Dexterity: The primary attribute for Demon Hunters and Monks. Dexterity increases Demon Hunter and Monk damage, and increases Dodge Chance for all classes.
• Intelligence: The primary attribute for Witch Doctors and Wizards. Intelligence increases Witch Doctor and Wizard damage, and increases resistances for all classes.
• Vitality: Increases Life, the amount of damage you can take before dying.
• Armor: Decreases the amount of damage you take from enemy attacks.
• Damage: Indicates the average amount of damage you deal with your attacks. Damage is derived from your equipped weapons, primary attribute, attack speed, and critical hit chance.

Primary class abilities provide an additional 1% damage per point.

Strength gives 1 point of armor per each point.

Every point of Intelligence grants .1 to all resistances.

As for knowing at a glance what items will be doing by adding stats, you can check the tool tip when comparing an item to what you have equipped, and it will show exact changes to all of your abilities, from damage to protection to health. Positive changes will be shown in green, and negative changes in red.