- Sending your password to your email in plain text. - Allowing user to set new password just by answering simple personal details like DOB, Zip code etc which might be known to a number of your friends and family members.
Often it is simple enough to implement a password reset functionality with a reset-link that contains a GUID which expires after a certain period of time. It could be more secure, but is a tradeoff between providing greater security and a longer reset process.
One of the major risks of websites with very naive reset procedures is that many people use the same password with multiple websites. So if a user's password gets compromised on a site, then the attacker can easily try those passwords with other services and gain easy access to user data.