|Assuming that this is your home or small office and that your router is the center of your network, meaning that you have no server of your own. if this is true then either you stumbled across a site that is trying to lure you into giving them your user name and password which others may be able to use against you, or your system is infected with malware and needs to be cleaned out.|
Avoid any site that asks you for anything that you have not initiated. You Can use a web based email and therefore create and enter the proper password to use it, you should never supply personal or secure information just because some site asks you for it.
Download, Update, and Run Malwarebytes (freeversion) to check for malware that got passed your antivirus program. During the install, make sure that you uncheck the 'Free Trial'. Remove all it finds and then post back the log that is created. I will recommend that JohnW look at it to see if there will be more needed to make sure that your machine is clean.
You have to be a little bit crazy to keep you from going insane.