What are the genetic changes that cause sickle cell anemia?

Mutations in the HBB gene cause sickle cell disease.

Hemoglobin consists of four protein subunits, typically, two subunits called alpha hemoglobin and two subunits called beta hemoglobin.
The HBB gene provides instructions for making beta hemoglobin. Various versions of beta hemoglobin result from different mutations in the HBB gene. One particular HBB mutation produces an abnormal version of beta hemoglobin known as hemoglobin S (HbS). Other mutations in the HBB gene lead to additional abnormal versions of beta hemoglobin such as hemoglobin C (HbC) and hemoglobin E (HbE).
In people with sickle cell disease, at least one of the beta hemoglobin subunits in hemoglobin is replaced with hemoglobin S.
More than 200 HBB mutations that cause beta thalassemia have been identified. Most of the mutations involve a change in a single DNA building block (nucleotide) within or near the HBB gene. Other mutations insert or delete a small number of nucleotides in the HBB gene.
http://ghr.nlm.nih.gov/condition=sicklecelldisease