Judging by your sig, it looks like you've at least started most of the classes, so if that means you plan on leveling the rest eventually, you might want to hold on to those aetherial items until the appropriate class is high enough to use them, that is, unless you also got a corresponding green-background piece that's even better for them.

Gear quality while leveling up is basically
(1) (pale grey background) standard gear like you get from quests, NPC vendors, or normal-quality crafting results.
(2) (pale grey background with sparkles) High-Quality results from crafting
(3) (pink background) aetherial gear dropped from dungeons (actually, this is a tie with High Quality gear that has materia added)
(4) (green background) unique dungeon dropped gear sets.

Once you get to endgame, there's more quality levels after that. (I think it's the blue background that comes next.) But by then, you'll mostly be looking at their Item Level.

If you don't plan on leveling the associated class, pink and green gear can be traded to your GC for a few GC seals. (And some green gear can be sold on the marketboard, but most cannot. I don't think any of the aetherial pink gear can be sold on the MB.)