During the first half of Act 2, you were offered (and may have picked up) one of the Herald skills: Herald of Ash, Thunder, Ice, Purity, or Agony.
For the most part, I've tried not to give exhaustive explanations of every skill in the game. Unfortunately, the Herald skills are such a common source of confusion for new players that they felt like they would be worth covering in a section of their own, because their mechanics are quite difficult to pick apart without a more thorough knowledge of PoE's mechanics.
(This page is very technical. If you're not using a Herald, or if you're not trying to use support gems to make it do things, you can skip this page, but Heralds are used by a large percentage of builds and are very easily misunderstood, so this is recommended reading at some point.)
Heralds are toggled skills that reserve Mana (leaving it unavailable for use), but unlike most such skills, they are not Auras - they only affect you, not your party members, and do not benefit from effects that specify 'Aura' in their language.
All Heralds have two separate effects.
The first effect is a passive buff on you (most of them just give added damage of the appropriate type). The language the skill uses is that it "Adds" damage, but this is kind of deceiving because the Herald itself is not dealing damage - in fact, what Heralds do is grant you an added-damage stat that then applies to the skills you use. This is important, because it means the Herald itself never does damage via this effect. In particular, the added damage alone does not count as a Hit for the Herald, even if the damage ends up being added to a Hit. This means that on-hit effects on Heralds do not simply trigger passively anytime you hit anything and that support gems on the Herald do not in general affect this passive added damage.
The second effect is a triggered effect that depends on you doing something while the Herald is active. This effect actually does deal damage directly (or summons a minion that does so), meaning that support gems and other effects on the Herald do apply. These effects have conditions, however, and only trigger when those conditions are met.
It is a very common mistake to link, say, Culling Strike to a Herald and expect it to apply on every hit, so I think it bears repeating: this does not work, because the Herald is not dealing damage itself through hits with other skills. It's giving you a stat that indirectly boosts those skills. Linked supports generally only apply to the secondary triggered effect. This can be useful, but it's mostly used in fairly specialized builds.
Yeah, it's a little confusing. It's self-consistent, and all of the above is perfectly within the normal conventions of Path of Exile's language, but if you're not well-versed in that language it's easy to misunderstand.
The mechanics of each Herald are individually a bit different, so it's worth talking about them in greater detail. All of the information above applies to all five Heralds, though some of it is repeated below to be absolutely clear.
Herald of Ash
Herald of Ash's passive effect adds a percentage of your Physical damage as Fire damage. This Physical damage can be from an Attack or a Spell (or even a Secondary source), although Damage over Time inherently cannot benefit from conversion (including "Added As" effects). See "How Damage Works" for more details on these mechanics.
In addition to this added damage, Herald of Ash also gives a passive bonus to your Spell Fire damage.
This passive effect does not ever Hit in its own right - it applies two stats to your character ("More Spell Fire Damage" and "% Physical Damage Added As Fire"), which then indirectly affect your damage.
Herald of Ash's triggered effect causes enemies you Overkill to inflict a Burn effect on nearby enemies for a percentage of the Overkill damage. As of this writing, this percentage is 25%, so if a monster had 100 Life left and you Hit them for 200 damage, you'd overkill them by 100 and inflict a Burn that deals 25 base damage to nearby enemies. Note that you must actually overkill with damage - effects like Culling Strike that kill a monster instantly will not trigger Herald of Ash. You can overkill with any type of damage, not just Fire damage. This Burn is a Damage Over Time effect, and neither its application nor its damage ticks count as a Hit, meaning that Herald of Ash is never capable of Hitting.
This burn effect breaks the normal damage modifier rules. Normally, a Burn effect - which deals Fire Damage over Time - would benefit from increases to Fire damage. However, Herald of Ash overrides this default behavior, and benefits only from Damage-over-Time-specific modifiers. This means it benefits only from increases to Burning damage (since "Burn" means "Fire Damage over Time") and Damage Over Time. In particular, unlike virtually all other effects, it does not benefit from generic Increased Damage. This is so that you cannot ever scale the Overkilling hit and the Burn using the same stat (you cannot "double dip").
As usual, the triggered effect benefits from any Support gems linked to Herald of Ash, subject to the unusual restrictions of modifiers listed above. In particular, it benefits from the Burning Damage Support and Increased Duration Support gems, which increase its damage and the length of the burn respectively.
Note that, as usual, a Burn is not the same as an Ignite. Herald of Ash cannot inflict Ignites in any way, because it is incapable of Hitting. Nor does this Burn count as an Ailment, which has a more specific meaning than just "negative effect"; it will not benefit from Ailment modifiers. For more on Ailments (and why this effect doesn't count as one), see their page in the Mechanics Reference.
Herald of Thunder
Herald of Thunder's passive effect grants you flat Added Lightning Damage to your Spells and Attacks. This is a stat on you, not damage done by the Herald itself, and therefore does not count as the Herald hitting (and is not affected by any Support gems linked to Herald of Thunder).
Herald of Thunder's triggered effect triggers when you kill a Shocked enemy (see the Mechanics Reference on Ailments for more on Shock). It creates a lightning storm centered on you that zaps nearby enemies, dealing Lightning damage, for a moderate period of time. This does count as a Hit from the Herald, and does apply on-hit effects and benefit from support gems linked to Herald of Thunder.
The damage dealt by the triggered effect is Lightning Secondary damage. Although the strikes hit a fixed area, which benefits from modifiers to Area of Effect, each strike is single-target; as a result, the strikes do not benefit from Area Damage. As Secondary damage, they do not benefit from increases to Spell Damage, even though Herald of Thunder has the Spell gem tag.
The damage from the triggered effect is specifically prevented from applying Shock, meaning that by default Herald of Thunder cannot sustain itself automatically. This is different from simply having no innate Chance to Shock - even if you do have Chance to Shock on your gear, Herald of Thunder's damage still cannot Shock.
Herald of Ice
Herald of Ice's passive effect grants you flat Added Cold Damage to your Spells and Attacks. This is a stat on you, not damage done by the Herald itself, and therefore does not count as the Herald hitting (and is not affected by any Support gems linked to Herald of Ice).
Herald of Ice's triggered effect triggers when you kill a Frozen (not Chilled) enemy (see the Mechanics Reference page on Ailments for more on Freeze). It causes the killed enemy to explode, dealing Cold damage in an area around it. This explosion does count as a Hit from Herald of Ice, and does benefit from any linked Support gems.
The triggered explosion damage is Cold Secondary Area Damage. As Secondary damage, it does not benefit from increases to Spell Damage, even though Herald of Ice has the Spell gem tag.
Herald of Purity
Herald of Purity's passive effect grants you flat Added Physical Damage to your Spells and Attacks. This is a stat on you, not damage done by the Herald itself, and therefore does not count as the Herald hitting (and is not affected by any Support gems linked to Herald of Purity).
Herald of Purity's triggered effect triggers when you kill an enemy (and has a chance to trigger when you hit a Rare or Unique enemy). It summons a Sentinel of Purity minion or, if you already have the maximum of 4 Sentinels of Purity, refreshes the duration and Life of one of them. Sentinels of Purity are Minions and obey the normal Minion mechanics, meaning that they benefit from Minion stats on you and from Support Gems linked to Herald of Purity, but not from your other stats. See the Mechanics Reference on Minions for more details.
Note that despite Herald of Purity having the Spell gem tag, no part of the skill benefits from increases to Spell damage in any way (although its added-damage effect may add damage to a skill that does benefit from it).
Herald of Agony
Herald of Agony's passive effect grants you extra Chance to Poison and More Poison Damage. Poison is an Ailment that can be applied through hits of Physical or Chaos damage, and Herald of Agony will generally speaking do nothing for you if you do not deal one of those types of damage. See the Mechanics Reference on Ailments for more on Poison and other Ailments.
Herald of Agony's triggered effect triggers when you Poison an enemy. Applying a Poison gives you a stack of an effect called Virulence; you can have up to 40 stacks of Virulence at a time. Stacks of Virulence decay over time, and the decay rate is faster the more Virulence you have. As soon as you gain your first stack of Virulence, you'll summon a single Agony Crawler, a Minion whose stats scale with the number of Virulence stacks you have. Agony Crawlers are Minions and obey the normal Minion mechanics, meaning that they benefit from Minion stats on you and from Support Gems linked to Herald of Agony, but not from your other stats. See the Mechanics Reference on Minions for more details.