The black vein that runs along the back of the shrimp is its digestive tract. While the vein is edible, it is full of unappetizing grit and is usually removed as part of the cooking process.
The process for de-veining a shrimp can be complicated and messy, but well worth it for both the taste and visual appeal of the final product. Kitchen shears, small knives and or de-veining tools made especially for the task can be used to de-vein shrimp. To de-vein, run a knife or other sharp tool along the back of the shrimp and pull the vein out. Shrimp are easier to de-vein before cooking.