Click the 'X' in upper right corner.

Yes but then someone else can just open it again, I want to log out completely and only log in with my password to avoid other people reading my emails.

You can set a primary password at settings>privacy&security. You must remember it, as there is no way to recover it if lost. The feature isn't perfect, as others could still read messages already downloaded, but there is a bug already filed to correct that. The only real safe way is to set up private accounts in Windows (if you use Windows).