Use fresh shrimp, or fresh frozen shrimp. The shell should have a crispness to it and shouldn't feel slimy (the older the shrimp, the soggier the shell).
To peel, place your thumb under the shell where the legs meet, and unwrap the shrimp one section at a time. To remove the vein, place a paper towel on a cutting board. Lay the shrimp on the paper towel with the inside curve facing you.
Using your finger, press gently on top of the shrimp. With a paring knife, make a gentle cut, only about an eighth-inch deep, along the outside curve of the shrimp, starting at the head and ending at the tail. With the tip of the knife, help the vein out of the flesh at one end of the shrimp. Then, with the tip of the knife, hold the vein down on the paper towel and pull the shrimp away.