What are they?
Gem coupons are items that can be traded in at the Event Master in Starglade for bound gems of various levels. The coupons themselves are bound (meaning they cannot be traded) and can be obtained a few different ways.
How do you use them?
To use coupons, talk to the Event Master in Starglade and choose the option "Gem Exchange." You will then be presented with a list of the available levels of gems you can trade for, from chipped to flawless. Select the level that matches the coupon you have. After you select the gem level it will ask you which type of gem you want. For chipped through normal gems, only ruby, sapphire, emerald, topaz, jade, amber, peridot, and garnet are avilable. For shining through flawless, the first page it shows you lists the same gems available for chipped through normal and you can click the option that says "Next Page" to see the additional gems: fire opal, citrine, kyanite, and pink diamond. The effects of the gems (though not the exact values of the effects for that specific level of gem) are listed next to the gem names. After selecting level (eg chipped, shining, etc) and type (eg ruby, emerald, kyanite, etc) you will be given a bound gem of that type and level. Remember that bound gems can only be synthed with other bound gems (and unbound with unbound) and that if you are synthing fragarach shards the 3 rubies and 3 topaz must either be all unbound or all bound.
How can you get them?
The first way is that they can be obtained from some early quests, though the quests usually give fairly low leveled coupons so they are not of much use as bound gems can only be synthed with other bound gems (more on why that's important in a moment). 2 more ways are to recieve them from the daily log in bonus or from events that occasionally give coupons (eg for a week you could recieve a scratched coupon for completing bath). The most common way, however, is by trading in with the gem coupon exchange.
Gem coupon exchange
At the Event Master you can trade in any type of scratched or shining gem for a coupon of the same level by spending 20 silver. Trade in procedure is very similar to the procedure outlined above for exchanging a coupon for a gem.