High Quality
A High Quality (HQ) item is an improved variant of an item which provides additional benefits. All gear, meals, medicines, and materials which can be crafted have HQ variants.
HQ items are highlighted with a golden shine on their icon and have a symbol in their names; they do not stack with normal quality (NQ) items of the same type. HQ items can be irreversibly downgraded to normal quality via an option in the right-click menu.
Obtaining
HQ items are not sold by vendors. Players can receive some HQ leveling gear as rewards for certain main scenario quests and side quests, often in the form of coffers. All other HQ items must be either crafted or bought from other players via the market board.
HQ items are crafted by increasing the "Quality" bar to its maximum before the craft is completed. This is done via "Touch" actions, which are affected by the Control attribute. (Any action listed on the Quality tab of the Calculations window while crafting is affected by Control.)
Effect on Item Stats
HQ gear and consumables have much better stats than their NQ counterparts. Specifically, NQ items have a 10% stat penalty compared to HQ items. This is particularly evident after level 50, when stats start to get large.
In content with other players, normal quality gear should never be used for battle after level 50. It is always better to use lower level gear, such as the preceding expansion's Blue rarity 
Poetics gear, than to use higher level but normal quality gear purchased from NPC vendors.
For endgame crafted gear and consumables, only HQ items should be used to participate in current high-end duties. The 10% stat penalty of NQ gear is comparable to using gear ~20 item levels lower.
Visually, HQ gear is exactly the same in every respect to Normal Quality gear, including the way it dyes. Therefore, for glamours the Normal Quality version has no downside.
Effect on Crafting
When used in crafting, HQ items provide up to a 50% bonus to the starting quality bar, depending on how many HQ materials are used. The degree to which 1 material increases the quality bar is different for each craft depending on the recipe.
Other Effects
HQ items provide double rewards when turned in to Grand Company Supply and Provisioning Missions. However, there is no bonus when turning in HQ items as expert deliveries or Ehcatl Nine deliveries.
History
During the early parts of Legacy, there were three tiers of high quality, called +1, +2, and +3. The +1 and +2 levels were removed early on, leaving only the +3. With A Realm Reborn, the +3 nomenclature was dropped.
Prior to Endwalker, gathered items also had high quality variants. For each gathering attempt, there was a chance that the item gathered would be HQ; the chance was influenced by the perception attribute and could be further modified through specific actions. As of patch 6.0, this mechanic was removed and replaced with the Gatherer's Boon effect, and only crafted items have high quality variants.