It's so easy! You just take a paring knife, score the apple into the checker pattern (this is the hardest part for me because my lines never end up straight). Then take the tip of the knife, hold it parallel to the apple, start in the corner of the peel you want to remove and carefully push/wiggle the knife around the square to remove the peel. A sharper knife is better.
To keep it from getting brown if you're going to save the apple for a few hours, dip it in cold water that has a bit of lemon (I use a few drops from a "plastic lemon").
Hope this helps!