Deveining shrimp, while partially a misnomer, is an important step in preparing perfect seafood. And it can be done in seconds!
Shrimp don't actually have veins because they have an open circulatory system; however, the process we call deveining does serve an important purpose. The first "vein" is the alimentary canal, or the "sand vein," and is where body wastes like sand pass through. You remove it, partly because it's unappetizing, but also so you don't bite down on the sand and grit.
The "white vein" on the inner crescent side of the shrimp is the blood vessel. It's white, rather than red, because the blood of shrimp is clear. There's no food-safety reason to remove this one, but you can if it seems more appetizing to you.
To devein your peeled shrimp, first start by keeping them in a bowl of ice water. It will keep them fresh while you work on the other shrimp. Holding the shrimp backside up, simply use a paring knife to cut a ¼-inch-deep slit down the back. Remove the "vein" with the tip of your knife, then rinse the shrimp in cold water.