What are Gems?

Guild Wars 2 will not have an monthly fee. There will be however micro-transactions, it will be possible to purchase stuff inside the game world with actual money. This is also found in its predecessor, Guild Wars (one). Unlike "free MMORPGs" players must still purchase the game first, just like any other game.

To buy stuff in the in-game gem shop you need to have gems. 

Gems are the in-game currency that you need if you want to buy stuff in the game shop inside Guild Wars 2. You can buy gems with real-life money and then spend these Gems in the game to get account services, decorative stuff and much much more, you can find a list of items you can buy with gems just down below on this page. I try to keep it up to date, if you find something that isn't correct please post a comment down on this page.
Gems can also be traded with other players for gold coins at the Trading Post (auction house)
There's also possible to buy Guild Wars 2 Gem Cards through retailers, (like the game time card for World of Warcraf, just that these are filled with gems instead) 

List of items you can buy with gems in Guild Wars 2 

Gem Store Home

Asuran Bank Portal
Dye Pack
Experience Boost
Black Lion Key

Style

Aviator Sunglasses Cook's Outfit Fine Transmutation Stone Inventor's Sunglasses Pirate Captain's Outfit Reading Glasses Ringmaster's Hat Sailor's Beanie Sport Sunglasses Top Hat Transmutation Stone Wide Rim Glasses Consumable Dye Pack Evon Gnashblade's Box o' Fun Mystic Forge Stone Black Lion Key Support Bag Slot Expansion Bank Access Express Black Lion Merchant Express Black Lion Trader Express Instant Repair Canister Black Lion Salvage Kit Revive Orb Account Bank Tab Expansion Character Slot Expansion World transfer Boosts Crafting Booster Experience Booster Karma Booster Killstreak Experience Booster Magic Find Booster Multi Booster Pack Mini-Pets Minis Pack