Pierce, Fork, & Chain
Shot Through the Heart, and You're to Chain
Pierce, Fork, and Chain are special effects that can be applied to Projectiles. Most Projectiles naturally disappear after striking a target and dealing damage, but these effects cause them to do something different. Note that some Projectiles already have one or more of these effects applied to them innately - for example, the Projectile from Freezing Pulse always Pierces as part of that skill's effect.
Some Projectiles apply their special effects on every hit: for example, the Wand Attack skill Kinetic Blast creates secondary explosions with each hit. Others apply their special effects only once when the projectile is destroyed: Fireball, for example, only explodes once at the end of its associated Projectile's life (although it still deals damage to targets it hits before that, just not to others around them). Check your particular skill for details, as this behavior varies unpredictably and is usually not noted in the skill's language.
These effects only apply to true Projectiles, and not just to anything that deals Projectile Damage. For example, the Bow skill Rain of Arrows deals Projectile Damage automatically (because any hit from a Bow skill does), but does not create true Projectiles and cannot benefit from these effects.
None of these effects work on terrain obstacles - you cannot Pierce an object or use it to trigger a Fork or Chain.
Pierce causes a projectile to continue moving past a target once it strikes. Most Pierce effects come with a maximum counter, as in "Projectiles Pierce X Additional Targets", although some Pierce effects are unlimited.
Projectiles that Pierce simply continue moving in the same direction they were already traveling.
Pierce takes priority over the other effects listed here. In particular, a Projectile that Pierces on a hit cannot Fork or Chain on that hit (although once it runs out of Pierces it may do so on the next hit). This means that, on Projectiles that always Pierce, Fork and Chain are useless.
Fork causes a Projectile to split into two pieces, each of which travels at a 60 degree angle from the Projectile's original trajectory, forming a sort of Y shape.
Unlike Pierce and Chain, Fork is a binary effect - a Projectile either can or cannot Fork. A Projectile can never fork more than once.
Fork occurs after Piercing is exhausted and before Chains can happen. Normally, a Projectile cannot Pierce after Forking, although at least one Unique item overrides this default behavior.
Chain causes a Projectile to "bounce" to another nearby target after it strikes one. It may or may not actually be able to hit that target (see below on "shotgunning"), but it will target them and travel in their direction, striking either them (if eligible) or potentially a different target if one happens to be in the way.
You can add as many Chains as you like to a projectile, and it will bounce once per Chain until its count is exhausted (at which point it will expire on the next impact).
Chain occurs after Fork and Pierce.
(Note: Some effects that are not Projectiles also 'Chain', although their mechanics are slightly different. These effects jump immediately to a new target on impact, and their Chain count can be increased by the same effects that would increase a Projectile's chain count [e.g. the Chain Support gem])
At one point in Path of Exile's history, it was possible to hit a single target with multiple Projectiles from the same skill use, or to have a single Projectile Chain back to a target multiple times. This effect - called "shotgunning" - caused a lot of balance issues.
In general, shotgunning is no longer possible. Projectiles skills keep a list of targets hit, shared among all the Projectiles created with each use of the skill and any Chained or Forked secondary projectiles they create, and will not strike a target already on the list. In general, a Projectile skill cannot hit a target more than once per skill use with its primary projectiles, even if they Fork, Chain, or shoot multiple projectiles.
Note, though, that the story is a bit more complicated than that in practice. While the direct hits of true Projectile skills themselves follow this rule, there are a number of ways in which a skill overall can hit a target more than once.
One common way to get around this restriction is to use Projectiles like Fireball that explode on impact. A single target can be damaged by more than one Fireball explosion, because the explosions are not hits from the Projectiles themselves. Similarly, some skills that Chain but are not true Projectiles, like the spell Arc, can chain to a target more than one time.
The full details of exactly what skills can and cannot use various secondary means to strike a target more than once are well beyond the scope of this guide. If in doubt, ask in in-game chat if a certain skill can "shotgun" or try Googling it (e.g. "Path of Exile Fireball explosions shotgun" will probably get you some useful results).