HQ (High-Quality) Items are always shown with a special symbol () next to their name. An item which is HQ will always sell to a vendor for more gil than the NQ (Normal-Quality) version will. Consequently, HQ items are usually rarer or harder to make than their NQ versions (there has to be some sort of downside to being better, after all!)
Weapons or Armor that are HQ will usually have better stats than their normal counterparts and are thus more sought out by players, and Consumables will have higher stats than their normal counterparts (such as a higher Vitality % earned on Meals.) Medicines which are HQ will always have a lower "recast" timer (the cooldown timer which determines when you can use another one) allowing you to use more of them during a fight, in addition to usually having a higher effect (whether it be recovering more HP via Potions with a higher cap, MP via Ethers, etc.) Almost all Weapons and Armor are created by other players; they cannot be found as drops from monsters, and can VERY rarely be found in chests. Most Consumables are made by players as well, but it is possible to get the HQ versions as drops, albeit at a lower drop rate than the NQ versions.
Turning in HQ items for Levequest rewards will DOUBLE the Gil () and Experience Points () you would receive for that turn-in. Pair this with the triple turn-in Leves (Sincerity and Charity), and you can potentially earn 6x the possible and that you would receive if you had just turned in the Leve once. Pre 2.1, this was the best way to level your Disciple of the Hand classes. To compensate for this, in Patch 2.1 Square-Enix buffed the amount of that non-Sincerity/Charity Leves will give, which currently makes them the best single option for experience-points-per-Leve-Allowance. They also require far less items to turn in, so unless you have an abundance of HQ items laying around that the Levequest requires... Most players will just pick the one that gives the highest amount of that a Leve will give, but it's recommended that you look closely at the rewards to decide which one you should do. As an example of a Triple Turn-in Leve versus a Single turn-in, let's look at two random examples: Ready for a Rematch vs The Arsenal of Theocracy.
Ready for a Rematch, a Triple Turn-in Charity Leve, gives 25,917 by default. Double that for a item, and it becomes 51,834 . Do that three times for 155,502 for one Levequest! The downside, is it requires 9 (NINE) Mythril Lances to do. Compare that to The Arsenal of Theocracy, which is a Delivery (Ingenuity) Levequest and awards 84,802 . Double that for and it becomes 169,604 . Since it's a single turn-in, it only requires 1 Cobalt Halberd. The downside to the Delivery Levequests, is they usually take a lot longer to turn in. The Arsenal of Theocracy, to use the example again, requires you to go from New Gridania to Whitebrim Front to turn in the Leve, which can take quite a while and cost a bit of Gil if you teleport there instead. So use your best judgement to determine which is the best Leve to use!
On this Wiki, both NQ and HQ items share the same page. The HQ stats are usually noted somewhere on that item's page (if that item can be HQ) so you usually won't have to look far to find out what they are. If you find an item on the Wiki that can be HQ, but doesn't have HQ stats, feel free to add them! It's incredibly easy to do so and only requires an account on the Wiki in order to edit. Just scroll down on the item to where the normal stat is listed (example: |Bonus Strength = +2) and underneath it add |Bonus Strength HQ = +3. Do the same for the other stats on the item, and don't forget Magic or Physical damage! ALL weapons for both Disciples of War and Disciples of Magic will have both stats on the weapon itself. The downside is that it won't be shown on The Lodestone or in-game. If you wish to add that stat (which you should,) you can either ask an Admin for help or look it up on one of the several Data-mining websites out there for the game! Once you've added everything (Don't worry about the vendor price. The Wiki automatically calculates the HQ price based off of the NQ vendor value, so just put that in if it's not there already) then Save the page to submit it, and you're done!