Don't count my old passwords as failed login attempts
Like most people, I have a lot of different passwords in my brain. While we really should have used a different system from passwords for web authentication, that's what we are stuck with now. A general good policy is to use the same password on sites you don't care much about and to use more specific passwords on sites where real harm could be done if somebody knows your password, such as your bank or email.
The problem is that over time you develop many passwords, and sometimes your browser does not remember them for you. So you go back to a site and try to log in, and you end up trying all your old common passwords. The problem: At many sites, if you enter the wrong password too many times, they lock you out, or at least slow you down. That's not unwise on their part, but a problem for you.
One solution: Sites can remember hashes of your old passwords. If you type in an old password, they can say, "No, that used to be your password but you have a new one now." And not count that as a failed attempt by a password cracker. This adds a very slight risk, in that it lets a very specific attacker who knows you super well get a few free hits if they have managed to learn your old passwords. But this risk is slight.
Of course they should store a hash of the password, not the actual password. No site should store the actual password. If a site can offer to mail you your old password rather than offering a link to reset the password, it means they are keeping it around. That's a security risk for you, and also means if you use a common password on such sites, they now know it and can log in as you on all the other sites you use that password at. Alas, it's hard to tell when creating an account whether a site stores the password or just a hash of it. (A hash allows them to tell if you have typed in the right password by comparing the hash of what you typed and the stored hash of the password back when you created it. A hash is one-way so they can't go from the hash to the actual password.) Alas, only a small minority of sites do this right.
This is just one of many things wrong with passwords. The only positive about them is you can keep a password entirely in your memory, and thus go to a random computer and login without anything but your brain. That is also part of what is wrong with them, in that others can do that too. And that the remote computers can quite easily be compromised and recording the password. The most secure systems use the combination of something in your memory and information in a device. Even today, though, people are wary of solutions that require them to carry a device. Pretty soon that will change and not having your device will be so rare as to not be an issue.