Two to three weeks ago, my husband's computer wasn't working, so I let him use my tablet. Since then we've been receiving each other's emails etc. He changed his password to try and fix the problem, but now it's not allowing me to sign in to YouTube etc. It tells me to sign in, but shows his Gmail account and asks for his password! What is up with that?
Quite simply, your husband didn't log out from gmail. This keeps him logged to any google owned website such as Youtube, Gmail and Google. Sign in with a different account as Steven proposes.