What are Mystic Coins used for? (And what are Zommoros's favorite trades?)

Take a look at the Wiki on the page Mystic Forge. It will tell you where the forge is, how it is used, and some predictable recipes. The forge lets you do two things. First you can create recipes to craft other items like consumables, weapons, or armor. Second, the forge can be used to create better crafting materials of one type higher.

The examples the wiki gives are as follows:

Common Crafting Material Promotion

Common crafting materials can be promoted to a small pile of higher tier materials with the following formula. The tier of dust needed is equivalent to the tier of the wanted product. So, if you are hoping to create Orichalcum, a tier 6 common material, you need to use Crystalline Dust, a tier 6 dust. If common crafting material B is one tier higher than common crafting material A, then:

10-50 Crafting Material B = 250 Common Crafting Material A (i.e. 250 Mithril Ores) + 1 Common Crafting Material B (i.e. 1 Orichalcum Ores) + 2-5 Dust A (i.e. 5 Pile of Crystalline Dust) + 5 Philosopher's stone

In that example: That recipe will produce 10-50 Orichalcum Ores

Armor boxes and armor satchels

Recipes can be created using the Mystic Forge. They will allow you to craft armor boxes and armor satchels, which will in turn give you every piece of an armor set.

Armor set recipe

Ingredients

5 Bottle of Elonian Wine

20 Arcane Crystal

50 Mystic Coin

1 Armor item

Mystic coins seem to be a common item needed for Food Trays, Weapons and Armor. Also look at this page for current recipe research.

You can see this in action with this video on how to make a Giant Chocolate Cake. Details are in the video summary.

To clarify what was outlined by Tapan Ze above, mystic coins appear to be used exclusively in Non-Random mystic forge recipes.

The most pertinent (and eponymous) are the Mystic Weapons (of which there is one of each), which have the following recipe:

Mystic Weapons

• Tier 6 Weapon Component A (i.e., 5x Orichalcum Blade)
• Tier 6 Weapon Component B (i.e., 5x Orichalcum Hilt)
• 10 Mystic Coins
• 1 Eldritch Scroll (50 skill points to buy)

Which are each level 80 exotics of whichever weapon you put in.

There are a great deal more recipes for level 80 exotic items, and mystic coins play a large part in them (and in great sums, too!)

This reddit thread has more (and is the source of the picture, above).