At this point, you've unlocked all the Skill Gems available in the game's main progression. Your last quest, from Lady Dialla, rewards your first exposure to Trigger Support Gems, one of the more interesting mechanics Path of Exile has to offer (and one responsible for a number of fascinating build options).
(Note that there are also three active skills - Riposte, Reckoning, and Vengeance - that are triggered by being hit or blocking. This page is not about those, since they don't by themselves trigger other skills, but they are mechanically speaking Trigger gems.)
Placing Traps, placing Totems, and laying Mines are not, in and of themselves, Spells, and therefore cannot be triggered. Nor can Totems, Traps, Mines, or Minions generally trigger Spells in their own right. However, a minion-summoning Spell can be triggered, allowing you to automatically summon minions.
Trigger Gem Mechanics
Trigger Support Gems allow you to 'trigger' (automatically cast) Spells. They do not trigger Attacks, Warcrys, or other types of Skill, only Spells. They have names of the form "Cast on <X> Support", and like all Support gems, must be linked to the gem you wish to trigger. A Spell linked to a Trigger gem cannot be cast itself, although a second copy of that Spell located elsewhere in your equipment still can be if needed.
Once a trigger's requirements (which vary from gem to gem) are met, the supported Spell will be cast automatically, without spending mana or interrupting the character's other actions. Trigger supports generally apply some form of cooldown to the triggered Spell, though the exact mechanics vary from gem to gem.
Vaal skills cannot be triggered, but their regular counterparts can, provided they meet the other requirements of the appropriate trigger gem.
Notable Trigger Gems
These four Trigger Supports are the most commonly used. The other three are rare or niche enough that we won't cover them here.
Cast When Damage Taken (Str/Int)
Cast When Damage Taken Support, often abbreviated to CWDT by the playerbase and build guides, triggers a supported Spell after you've taken a certain amount of damage. It is by far the most important of the Trigger supports, and is utilized in some form by virtually all builds.
Each linked Spell to CWDT tracks damage taken separately, and it will trigger as soon as the threshold of damage listed on the CWDT gem is met. The damage does not all have to be done in one hit. The Spell will be cast at the enemy that caused the damage. Once a Spell is triggered, it is put on cooldown for a brief period and will not trigger or track incoming damage during that time (other Spells of the same name are also prevented from triggering during this time).
CWDT applies a substantial damage penalty to triggered skills. As a result, it usually used more to apply utility effects than to deal damage in its own right.
Higher levels of CWDT require more damage to trigger, making this one of the few gems you may not wish to level up even when able to do so.
Important: Cast When Damage Taken, unlike other supports, has a cap on the required level of the supported Spell: at Level 1 CWDT, supported Spells cannot have a level requirement over 38. If the Spell exceeds this requirement, it will not be supported at all by CWDT - it won't be triggered, nor will self-casting the skill be disabled. The cap does not apply to any other Support gems, which can be leveled freely.
Cast While Channelling (Int/Dex)
Cast While Channelling Support, often abbreviated CWC, must be linked both to a regular Spell and to a Skill with the Channelling tag. The Channeled skill can be either a Spell or an Attack (CWC works with either) but the triggered skill must be a Spell.
While using supported Channelled skills, the supported regular Spell will be cast periodically. For example, if you linked CWC to the Channelled Skill Scorching Ray and to the Spell Skill Fireball, channelling the Scorching Ray would cause you to shoot Fireballs periodically while you do so.
The rate at which CWC fires supported Spells does not depend on the cast time of the Spell in question. As a result, CWC is best used with Spells that have long cast times ('long' here is something close to a full second). You will generally want to find a Channelled skill and a Spell that can benefit from the same Support Gems, so as to scale the damage of both, although this is not universal to CWC-centric builds.
Cast On Critical Strike (Dex/Int)
Cast On Critical Strike Support, often abbreviated COC, must be linked to both an Attack and a Spell.
When the supported Attack successfully lands a Critical Strike (see the Mechanics Reference for details on Critical Strikes), the supported Spell will automatically trigger. That spell cannot be triggered again for a short time. This cooldown is per-spell; an Attack can trigger more than one Spell provided they're all linked to the same COC Support.
Important: It is a common mistake for players to assume that this causes supported Spells to trigger on any Critical Strike. However, it works only on Critical Strikes with Attacks in the same link as COC and the triggered Spell. A spell not linked to both COC and the critting Attack will not be triggered.
COC is, as you might imagine, used mostly by builds that scale Critical Strike, since that both (a) allows it to trigger in the first place and in some builds (b) scales the damage of both the Attack and the triggered Spell at once.
Curse On Hit (Int)
Curse on Hit, or COH for short, must be linked to both a Curse skill and to a skill that can Hit (say, an Attack or a Spell, but note that Damage over Time effects do not Hit in themselves even when applied by Attacks or Spells).
When the supported skill Hits, it will apply the supported Curse's effect to all enemies Hit. Note that it does not actually cast the relevant Curse, and will not affect all enemies in an area unless those enemies were actually hit by the triggering Hit.
The triggering skill can, itself, be triggered. For example, it is possible to link Cast When Damage Taken, a Spell that Hits, Curse on Hit, and a Curse together in a four-link item. When you take damage, the Spell casts, which Hits enemies, which applies the Curse. (However, because any one skill can only be supported by one gem that might trigger it, the Curse in this link must be leveled up to the point that it is ineligible to be supported by the Cast When Damage Taken gem.)