For example, say your password is "Superman". Amazon's login would accept "superman", "SuPeRmAn", "SUPERMANISCOOL", or "superman12345". Again, this supposedly only affects older passwords, but a precise timeframe isn't known. Two commenters claim their 2008 and 2009 accounts are affected, while others say they haven't changed their password in six years and everything is fine. For what it's worth, the flaw doesn't work on my 2004 account.
An Amazon employee posting on Reddit said the company is aware of the issue and it's being addressed. In the meantime, you can mitigate the problem by going through Amazon's password change procedure. Your "new" password can be identical to your old one, but it will be stored with improved security.