The below information comes from http://www.geocities.com/pelionature/mythology_Arachne.htm.
 
 

There once was a girl named Arachne. She was an extremely skillful weaver.  (Weavers are people who make colorful rugs and cloth by spinning threads and cloth-strips together.)  The tapestries and cloths she made were so fine and delicate, that people  marveled at her creations.
 

Arachne was too proud of herself, however, and she became arrogant.  She told people that no one was as good a weaver as her--even Athena, a Greek goddess who was known as a wonderful weaver.  
 

The goddess Athena was angered by Arachne's arrogance, so she came to the girl disguised as an old woman and warned her not to say bad things about the gods.   Instead of listening to the advice, Arachne told her to get lost.

Furious, Athena took off her disguise and challenged Arachne to a contest to see who was the best weaver.
 

Athena began weaving, and created a beautiful cloth that had pictures on it.  It showed scenes of people being punished for doing bad things to the Greek gods.
 

Then Arachne began weaving.  In an attempt to be funny, she created a cloth that showed the Greek gods doing bad and embarrassing things.
 

When Athena saw this, she got angry.  She grabbed Arachne's cloth and tried to find mistakes in it.  There were none.  Arachne was truly a great weaver.  Furious with jealousy, and angered by the pictures Arachne made for her cloth, the goddess tore the cloth to shreds and hit Arachne across the forehead.
 

Arachne was ashamed and embarrassed by being insulted in front of so many people.  She left to kill herself by hanging herself in a tree, but Athena didn't let her die. Instead, Athena turned Arachne into a spider and sentenced her and her descendents to a life of eternal weaving while being suspended from a thread.

People still remember Arachne to this day, as they have given her name to a a certain kind of creature known as arachnids, or spiders.

Why do you think Arachne was turned into a spider?  Why might it have to do with hanging herself?