I'm using Windows 8.1 Mail app for much time, and now decided to switch to Outlook.

So there's one problem: when I'm adding an account, the window "Windows Security" with fields for credentials appears infinitely. I'm sure i'm using right password, 'cause it works with all other services and even I've changed it in hope this should help, but it didn't.

I don't have any more thoughts how to make it work. Can anybody help me what's the reason?

So which account type and provider are you using?

When using an Outlook.com or Gmail account with 2-factor auth enabled, you'll need to use an App Password instead of your regular password.
See: Outlook and Two-Step Authentication for Outlook.com and Hotmail account
and: Outlook and 2-Step Verification for Gmail accounts

When you are using Gmail with 2-factor auth disabled, you'll need to adjust a security setting to be able to connect via a mail client. See: http://www.howto-outlook.com/news/gmailoauth.htm