What benefits do the different gems provide? How does quality boost it?

What benefits do the amethyst, emerald, topaz and ruby provide when socketed? How is this affected by the quality of the gem?

Solution 1:

The complete details for each gem type and quality level appear on the official guide: http://eu.battle.net/d3/en/item/gem/

In short:

           Helm       Weapon            Other
Amethyst   +%Life     +Life per hit     +Vitality
Emerald    +Gold      +Critical Damage  +Dexterity
Ruby       +XP        +Damage           +Strength
Topaz      +MF        +Thorns           +Intelligence

+MF means Magic Find.

Solution 2:

This website has a pretty good guide to gems.

1. Amethysts give you life percentage when socketed into helms. When socketed in other equipment, amethysts give you Vitality. When socketed into weapons it gives lifesteal.

2. Rubies increases the experience you gain per kill when socketed into helms. When socketed in other equipment, rubies give you Strength. When socketed into weapons gives +max/mix damage.

3. Emeralds give you extra gold from monsters when socketed into helms. When socketed in other equipment, emeralds give you Dexterity. When socketed into weapons it gives crit chance.

4. Topazes give you better change of finding magical items when socketed into helms. When socketed in other equipment, topazes give you Intelligence. When socketed into weapons it returns damage taken.