MacOS – Mac only approves password on second entry

If you don't want to try one more time to reset the password, you might need to contact Apple's support to have them reset your account.

It's entirely possible that the account is either locked or someone else is trying to reset the password (or far more likely another device you have stored the old password keeps repeatedly trying to enter that old password and is causing the account to get locked.)

You can try logging into https://appleid.apple.com from a web browser and see if you really have the correct password. Once you're certain the account isn't locked, you can then sign out of all your OS X and iOS devices to ensure that the old password is cleared and not stored incorrectly.

If your account is locked due to too many incorrect attempts - you will need to contact Apple to get things sorted.

You need to open the Users & Groups section of the system preferences and check what users are available. Is there one with admin privileges? Use that for the installation dialog. The inconvenience will be that in those dialogs you will need to enter the admin user name in addition to the password.

If there is no admin user, you could try to enable a root account. However, this needs admin privileges to be set up. Check if the padlock icon in the lower left corner of the Users & Groups section is closed (needs password, click on icon to open the password dialog), or open (you are free to add accounts). If the padlock icon is closed and you do not have any working admin user - password combination, I recommend to re-install OS 10.9 and to add the admin user at the appropriate prompt in the installation process. I hope this won't be needed - this should really be your last resort.

For Apple's explanation of the root account setup, see this article.