there are two kinds of ways password can be kept
this page uses both of them as a way of testing things
if you have autocomplete active (IE, Tools, Internet Options, Content, Settings), the first will ask you for passwords, if not, or if you've already clicked not to save the password once, it won't ask you no more
you can also delete all browsing history, and delete HKCU\Software\Microsoft\Internet Explorer\IntelliForms\Storage1, Storage2, etc
maybe doing some backups before, exporting those keys
The second can be a site-related issue, or IE doesn't qualify the page as a login form for some reason. This second issue of commonly found in SharePoint Sites, both in IE and SharePoint Designer. They ask for the password, but it won't remember it when you tell it to.
One way to fix this is set automatic authentication, where it will use the current windows login data as authentication for the site.
To do this, open IE options, and add the site to "Local intranet" in secutity". In my case, I had to remove it first from the Trusted Zones. Then click Customize, and select "Automatic logon with current user name and password".
Thanks to this, I no longer have to keep writing down the password each time I close Internet Explorer and open my SharePoint sites again. Of course, Firefox does save all passwords when we ask it to.