This myth got its start a number of years ago when medical researchers found elevated levels of aluminum in diseased tissue from the brains of Alzheimer's patients.
One logical possibility (but not the only one) was that the raised aluminum level was responsible for causing the disease. Get exposed to too much aluminum, from your job perhaps or your cookware, and you would have a better chance of coming down with this awful disease. People started avoiding aluminum cookware, and some still are - unnecessarily it turns out.
Subsequent research has failed to show any connection between aluminum exposure and Alzheimer's, and it is believed that the elevated aluminum in the brains of Alzheimer's patients is a result of the disease process. In other words, high aluminum levels do not cause Alzheimer's, but rather Alzheimer's causes high aluminum levels.
Source: Alzheimer's Society