For the book the stories are saying Aes Sedai are witches and are sometimes called the witches of Tar Valon. When you are a teenager and you have the one power in you you are taken to the white tower to train. On your way there you train every night.
When you get there you are a novice and you are not allowed out of the tower. When you are powerful enough you can be raised to the accepted.
To do that you have to face your fears three times. Then you get to wear a white dress with seven bands of colour around the waist. You also get a golden ring and on it there is a golden serpent biting its tail. Then you live in the accepted quarters. When you are powerful enough you are raised to Aes Sedai.
To do that you have to channel under pressure, recieve the shawl, take oaths and choose your ajah.