Boil 1 ½ cups water in a medium saucepan and add shrimp. Reduce heat to medium and cook for 2 ½ to 3 minutes (stirring occasionally), or until shrimp are cooked and have just turned pink (do not overcook).
Drain shrimp in a colander, rinse with cold water and allow to cool.
While shrimp are cooling, whisk together mayonnaise, lemon juice, dill weed and Old Bay Seasoning in a medium bowl. Cover and refrigerate until shrimp are ready to be added.
Peel and devein shrimp and cut in half (or chop, depending upon preference).
Add shrimp, red onions and celery to mayonnaise mixture and stir well to coat. Cover and refrigerate for at least 1 hour before serving.