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What happened is that you were using Local account. When your friend installed Skype, probably they used the "Modern" / Metro app of Skype. A pop-up windows asked you/your friend to change from Local account to Microsoft. And here is the big mistake - this was not necessary and this is where the problems started. Now your friend is actually the owner of the account. Your local account is migrated to the Microsoft account. More info:
Since your friend has @ gmail.com account, this means he/she has both Google account (gmail.com) and Microsoft account (gmail.com). When entering the login password, ensure your computer is connected to the Internet (e.g. cable or wi-fi connected prior to login screen).
Your friend must be careful when they tell you their password - it must be the Skype one/Microsoft account one - not the gmail.com (Google account) one. If they are sure, then they probably are not telling the truth about the password.
Something else you can try:
System Restore - restore back to the state before Skype was used.
Restart your computer. As soon as you notice the HP logo, keep hitting the Esc button (like tap-tap-tap).
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