Diamonds are currency used for better quality gear and consumables like Scrolls of Identification. This is also only currency of auction house. You can pay with diamonds for teleportation scrolls, better gear, stronger companions and faster mounts.
Shops where main (in most cases only) currency are Astral Diamonds.
Auction House (AH) is players market and only currency here are diamonds.
Auction Broker offers help with equipment search for diamonds.
In Zen Exchange panel from quick menu you can exchange diamonds for zen.
Main source of astral diamonds are daily quests and invocation system.
In most cases you will get Rough Astral Diamonds as a reward for quests and other tasks, those have to be refined into normal diamonds in Riches tab of your inventory. You can only refine set amount of Rough Astral Diamonds daily.