Hold the ginger on one end, and with the edge of a small metal spoon, begin scraping the peel with light to moderate force. Continue this until all of the skin has been removed.
The fibers of a ginger run lengthwise towards the root, so it's important to cut against the grains of fibers. Lay the peeled ginger down on a cutting board, and with a sharp knife cut coins to the desired thickness.
For minced ginger, make thinner coin slices, place two or three coins on top of each other and cut them into matchsticks. Then line up the matchsticks and make cross cuts.
If your recipe calls for ginger paste, use a microplane to run the ginger against the plane and make a paste.
Notes
Tip: Don't try and use a knife to peel your fresh ginger. You'll waste more than you'll gain. A spoon is really the way to go!