A process by which a fruit of some kind (unwaxed) is shaved along its peel. The resulting shavings are then used for a variety of purposes. Coarser zest can be used to add textures to different foods. Finer zests can be used to add a bit of flavor. For that latter purpose, this is especially useful if the preparation process could break down the citric acid and other compounds that produce the desired flavors.
When trying to produce a finer zest (perfect for baking, marinades, sauces, etc.), you may want to use a zester of some kind. If you are wanting to use a microplane, then you have the right idea. Now, if you want a coarser zest, you should probably use a grater to produce the longer stretches of zest that you want.