Attributes
Attributes determine your character's statistics in combat, gathering, and crafting. Different races start off with very slightly different base attributes. Players commonly refer to these as "main stats" or "core stats."
Strength
Affects physical damage dealt by 
Gladiator's arms, 
Marauder's arms, 
Dark Knight's arms, 
Gunbreaker's arms, 
Pugilist's arms, 
Lancer's arms, 
Samurai's arms, 
Reaper's arms, 
Thaumaturge's arms, 
Arcanist's arms, 
Red Mage's arms, 
Pictomancer's arms, 
Conjurer's arms, 
Astrologian's arms, 
Sage's arms, and 
Blue Mage's arms.
Dexterity
Affects physical ranged damage, and damage dealt by 
Rogue's arms and 
Viper's arms.
Vitality
Affects maximum HP.
Intelligence
Affects attack magic potency when role is 
DPS.
Mind
Affects healing magic potency. Also affects attack magic potency when role is 
Healer.
Points
HP
HP are your health points. As you take damage, your HP will decrease. You are Knocked Out when your HP reaches 0. After you are KO'd, you must be resurrected by another player or return to your Home Point/the start of the duty.
Players may temporarily increase their Vitality by consuming Meals, effectively increasing maximum HP.
MP
MP, or magic points, are required to cast spells and use certain magical abilities. If the MP cost of a spell is greater than your remaining MP, you cannot cast that spell. MP slowly regenerates over time in and out of combat.
MP formerly varied based on Piety, much like HP does with Vitality. As of Patch 5.0 (Shadowbringers), the MP cap for all levels was changed to 10,000, and MP costs for all actions became fixed values.
All Disciples of Magic (
Magical Ranged DPS and Healer) have actions that require MP. Additionally, two 
Tank Jobs also have actions utilizing MP: 
Paladin and Dark Knight. For the tanks, spells are scaled on strength rather than intelligence or mind.
Meldable Stats
These are commonly referred to by players as "substats."
Critical Hit
Affects the amount of physical and magical damage dealt, as well as HP restored. The higher the value, the higher the frequency with which your hits will be critical AND the higher the potency of critical hits. By default, the chance of a critical hit is 5%, with a damage increase of 40%.
When under an effect that increases the chance of a Critical Hit, actions that are guaranteed to deal a Critical Hit will deal increased damage, according to the Critical Hit rate (as of Patch 6.2).
If an attack is both a Critical Hit and a Direct Hit, it is considered a Critical Direct Hit, and the damage boosts of the two effects are multiplied by each other.
Heals can Critical Hit, but they cannot be Direct Hits.
Because Critical Hit acts to boost both the frequency and power of criticals, it is widely considered the most important secondary attribute for increasing DPS and as such is often prioritized for materia melding.
Determination
Affects the amount of damage dealt by both physical and magic attacks, as well as the amount of HP restored by healing spells.
Because Determination has a lower coefficient than either Direct Hit Rate or Critical Hit, it is generally considered the weakest of the three DPS substats.
Determination is somewhat more consistent as its effect is always active, and it is slightly more flexible than Direct Hit Rate as Determination also improves healing.
While the in-game tooltip specifies that determination only affect healing spells (i.e. global cooldown actions), it also boosts the healing from abilities (i.e., off-global cooldown actions).
Direct Hit Rate
Affects the rate at which your physical and magic attacks land Direct Hits, dealing 25% more damage than normal hits (this percentage of extra damage is fixed). The higher the value, the higher the frequency with which your hits will be direct.
When under an effect that increases the chance of a Direct Hit, actions that are guaranteed to deal a Direct Hit will deal increased damage, according to the Direct Hit Rate (as of Patch 6.2).
If an attack is both a Critical Hit and a Direct Hit, it is considered a Critical Direct Hit, and the damage boosts of the two effects are multiplied by each other.
Heals can Critical Hit, but they cannot be Direct Hits.
Skill Speed
Affects both the casting and recast timers, as well as the damage over time potency for weaponskills and auto-attacks (but not abilities or spells). The higher the value, the shorter the timers/higher the potency.
Spell Speed
Affects both the casting and recast timers for spells (but not abilities or weaponskills). The higher the value, the shorter the timers. Also affects a spell's damage over time or healing over time potency.
Role
Tenacity
Affects the amount of physical and magical damage dealt and received, as well as HP restored. The higher the value, the more damage dealt, the more HP restored, and the less damage taken. Only applicable when role is Tank.
Generally considered an undesirable attribute due to its relative lack of impact compared to substats such as Critical Hit and Direct Hit Rate, so materia melds and best-in-slot gear avoid tenacity when possible.
Piety
Affects MP regeneration during combat. Regeneration rate is determined by piety. Only applicable when in battle and role is Healer.
Due to the wealth of MP-free healing actions and availability of other MP management actions that high-level healers have, piety is generally minimized when possible in favor of DPS-increasing substats such as Critical Hit. However, some amount of piety is helpful when progressing high-end duties. Consumable MP restoration may also be used to compensate for low Piety.
Other Properties
Attack Power
Affects amount of damage dealt by physical attacks. The higher the value, the more damage dealt. Attack Power is exclusive to Weapons.
Attack Magic Potency
Affects the amount of damage dealt by magic attacks. The higher the value, the more damage dealt. Attack Magic Potency is exclusive to Weapons.
Healing Magic Potency
Affects the amount of HP restored via healing magic. The higher the value, the more HP restored. Healing Magic Potency is exclusive to Weapons.
Defense
Affects the amount of damage taken by physical attacks. The higher the value, the less damage taken. Defense is exclusive to Armor. All Accessories provide only 0 or 1 Defense.
Magic Defense
Affects the amount of damage taken by magic attacks. The higher the value, the less damage taken. Magic Defense is exclusive to Armor. All Accessories provide only 0 or 1 Magic Defense.
Crafting
Craftsmanship
Affects the amount of progress achieved in a single synthesis step. Some recipes need a minimum amount of craftsmanship either for Quick Synthesis or to even attempt to craft.
Control
The ingame tooltip says "Affects the amount of quality improved in a single synthesis step."
However, this is misleading, since Quality is improved by Touch, and Progress is what is increased by Synthesis.
Control increases the amount of quality added by all Touch actions, making it easier to increase your HQ chance.
Some recipes need a minimum amount of control either for Quick Synthesis or to even attempt to craft.
CP
CP, or crafting points, are required to use some of your crafting actions. When you have a crafting class equipped, CP bar replaces your MP bar. CP regenerates fully after you complete a craft, but does not regenerate during crafting actions, though there are a few actions that can restore CP during a craft.
Generally considered the most important crafting attribute, because a minimum amount of CP is required to complete a given crafting rotation. That being said, CP is a "threshold" thing where you need enough for the rotation, but any extra is worthless.
Gathering
Gathering
Affects the rate at which items are successfully gathered. For 
miner and 
botanist, too little Gathering can result in a miss instead of receiving an item. The success rate of gathering an item increases up to 100%, and is indicated next to the item in a node, based on the gathering attribute. Affects the yield increase (+1 to +3) of 
Bountiful Harvest / 
Bountiful Harvest II / 
Bountiful Yield / 
Bountiful Yield II depending on the node. Reaching specific gathering thresholds for a given node will randomly result a bonus in yield. For 
collectables, higher gathering will result in a higher collectability increase when using 
Scour and a higher chance node integrity will not be reduced (depending on the node) when using 
Meticulous Prospector or Meticulous Woodsman.
For 
fisher, a minimum gathering stat is needed to catch any fish. However, for rod fishing, the success rate of catching a fish may not become 100% depending on the fish. This is especially true for big fish, which have a noticeable failure rate regardless of stats. Increasing gathering above the minimum requirement does not appear to significantly increase the success rate of catching a given fish (e.g., it is possible, although unlikely, for a level 1 fish to escape the hook with endgame gear). Gathering also affects the yield increase from 
Double Hook and 
Triple Hook. For spearfishing, increasing gathering will increase the hitbox size of fish up to a certain length, making them easier to catch. Below the minimum gathering stat, the fish will not have a hitbox and not be possible to catch.
Perception
Affects the rate at which Gatherer's Boon is triggered for Miner and Botanist. Additionally increases the collectability increase provided by 
Scrutiny when gathering collectables. Certain items in gathering nodes require a minimum perception stat to be eligible for gathering. Reaching specific perception thresholds for a given node will randomly result in a bonus in Gatherer's Boon rate (up to 100%).
For Fisher, increases the chance of gathering large-sized fish when fishing or spearfishing. Will also result in a higher chance of gathering a large-sized fish upon using 
Nature's Bounty.
GP
GP, or gathering points, are required to use some of your gathering actions. While you have a gathering class equipped, GP bar replaces your MP bar. GP slowly regenerates when outside of gathering actions, but does not regenerate during gathering actions. Reaching specific GP thresholds for a given node will randomly result in a node integrity bonus, allowing for more gathering attempts.