Finally, a question we can answer with confidence!
Zest is the outermost portion of the peel of limes, lemons, oranges and other citrus fruits - but only the colored portion. The white part just underneath is called the pith, and is bitter. The zest contains aromatic oils that really boost the flavor in certain recipes.
When you see zest specified in a recipe, the author is asking you to include tiny little filaments of the peel, as opposed to larger strips. There are special kitchen tools on the market called zesters to help you scrape off the zest without digging into the pith. The most common have a row of stiff metal rings that peel five or six threads of zest at a time. The current darling of the cooking world, though, is a Microplane Grater/Zester based on a woodworker’s rasp, that has found a second career as a super-hero zester. (Go to any gathering of food professionals and just hear them gush about their extra-fine zest - it’s a little much for our tastes, but it is a great tool.)
If you don’t have space in your kitchen for one more gadget, you can also use a vegetable peeler or sharp knife to cut wider strips of zest, which you can then cut into narrow strips with your knife. But remember, avoid the pith. Better to have no zest in your recipe than zest and pith! (Say that five times fast….)