Removed gems can be traded and sold with other players or even combined with 2 other gems of the same color and quality to make the next tier of quality. You can place gems into sockets and even remove them. Head, chest, wrists, waist, amulets, shields and weapons can be socketed with gems. Gems are inserted into their gear just like in Diablo 2 and in World of Warcraft.
In Diablo 3, just about everyone wants or needs gems to enhance the power of their gear. If you want to make gold, join the Diablo 3 Secrets community forum to find out how people are doing it right now. This is true for the first 8 tiers of gems right now. If you have 3 Flawless Emeralds and want 1 Perfect Emerald just sell the 3 flawless on the auction house and then buy 1 emerald with the cash you get. You will actually lose gold combining gems (except some high level rubies and other rare ones). Warning: It is currently not possible to make gold combining most gems. You will level these two Artisans up as you progress through Diablo 3 and they will allow you to craft an expanding assortment of equipment and other bonus items.
When you play Diablo 3 you will meet your very own Artisan Jeweler (named Covetous Shen) who you can work with and train to higher levels.
Jewelcrafting in Diablo 3 is all about gems, tons of gems.