I've got Nokia Lumia 636 with Windows Phone 8.1. One of these days it stopped to log in to Skype. I tried to log in with Skype name, and with Microsoft account. Both methods fail. Although, I use Gmail's mailbox for the latter, if that matters. I can log in with both credentials on login.live.com. Why is that? Is there a way to make it work on my phone?
Skype is no longer supported on Windows Phone 8.1 as of July 2017. Your options pretty much boil down to the following:
Buy a newer Windows phone running Windows 10 Mobile
Use another instant messaging app (but note that those might also end WP8.1 support at any time; Facebook for example has already started doing so)
Switch to Android or iPhone