Bejeweling is a skill that is gained in the Uncharted Waters expansion pack. It can be gained only by gaining access to Utopia Town on Paradise Island.
You do not need to own the expansion pack to use a bejewel but you do need to own the expansion pack to make them. This means people with the expansion pack have a good way of making extra money by selling bejewels to other players.
To fit an item you have made with bejeweling skill simply use the item from your inventory, it will then provide a list of the compatible items in your inventory that you can fit the bejewel into.
Simply select the item you wish to fit the bejewel into and then click the Install button.
You may not fit a patch made with Patchcrafting and a bejewel into the same item, although you can remove a patch/bejewel fitted in an item to fit a different one.