Why does Ganesha have the body of a human and the head of an elephant?

There are many stories which tell how Ganesha came to be.

Here is a popular one:

The Goddess Parvati was bathing and wanted some privacy. She scrubbed some sandlewood paste off her body and, out of it, created the figure of a boy. She told him to guard the entrance to her rooms while she bathed.

Soon after this, her husband, Lord Shiva,  returned home looking for his wife. The young boy blocked his way and, because he did not know who he was, Shiva became extremely angry and in the battle that followed, Shiva cut off the boy's head.

When she returned from her bath, Parvati saw she headless son and threatened to destroy the heavens and the earth, so great was her sorrow.

Shiva calmed her down and told his followers to bring the head of the first living thing that they met. The first creature they found was an elephant. They cut off its head and it was placed on the body of Parvati's son. Shiva breathed life into him and Parvati embraced her son.

Shiva gave him the name Ganesha.