Skill Gems I
Skill Gems I: The Basics
In Path of Exile, all of your skills (other than your default attack) come from items called 'skill gems'. Skill gems must be socketed into your equipped weapons and armour in order for you to be able to use the skill they grant. It doesn't matter which item the gem is socketed into, but the socket has to match the colour of the gem. In other words, red skill gems can go in red sockets, and can't go in blue or green sockets. Support gems work very similarly to skill gems, except they don't grant you a skill by themselves. Instead, support gems can augment the skill gems they are linked to. Not all skill and support gems work together; You can't use a 'Melee Physical Damage Support' with a spell, for example.
For a support gem to properly augment a skill gem, both the skill and support gems must be socketed in linked sockets. If they aren't linked, the support will have no effect. You can tell whether or not your support gem is affecting your skill gem by looking at the on-screen skill icon. Properly linked support gems add a little letter to the skill icon on the user interface. Most support gems also change the mana cost of the skills they're supporting.
You can add or remove your skill and support gems at any time, so don't be afraid to experiment with strange gem combinations.
Skill Gems II
Skill Gems II: Requirements and Restrictions
Support and skill gems, like other items, have level and attribute requirements. Unlike other items, gems can level up, and when a gem levels up, its level and attribute requirements can change. If you level a gem past the point that you can use it, or if your attributes change and disable a gem, the entire item the gem is socketed in will also be disabled. That means it's worth checking whether or not you should level up gems that don't require your primary attributes. Sometimes it's better not to level up a gem!
You may find that you want to use the same support gem for multiple skills. Simply socket that support in the same item as the skills you wish to support and make sure they are all linked. The support gem doesn't have to be adjacent to the skill gems you want to support; they just have to all be part of the same group of linked sockets. You may also find that you want the same skill to be supported by multiple support gems. The requirements for this are the same.
Supports That Change Skill Source:
Some support gems will significantly change the way the supported skills behave. A spell supported by the 'Spell Totem' support will now summon a totem to cast the spell for you, and you won't be able to cast the spell yourself. The same is true for the 'Trap' and 'Remote Mine' supports.
Some support gems can be used to trigger a skill automatically. The 'Cast on Critical Strike Support' gem casts the spells it is supporting whenever you get a critical strike with an attack it is also supporting. Any spell supported this way can only be triggered and can no longer be cast manually, similar to how the Spell Totem Support works. Skills being supported by multiple trigger gems won't work at all. Keep this in mind when deciding which skills and supports you want to use.