Being stateless does not mean that NO information is present on the server, it just means that the request performed by the client contains all the needed information for the server to handle the request.
When you log on to a server, that server can send you a session token so it can persist information across requests. All the server needs is for you to send the token back to the server on every subsequent request and it can associate the token with the information that was persisted. This is indeed the way the HTTP-part of the web works.
Compare this to what FTP does: when you connect, the server sets certain variables for you. For example, it 'knows' what your current working directory is and what file transfer mode (ASCII, Binary...) you will use. Uploading a file to an FTP server does not involve you sending the file, along with what the current directory should be, what the transfer mode is, etc... The server 'knows' this because this information is associated with your session.
As for your changing a PC: when you log into a website, it can log what page you are on as you visit them and persist this to a database. If you explicitly log out this information is removed from the database. As long as you don't do that, logging into the website from another client simply puts you back onto the last page you were: the application identifies you by your username, fetches the last page from the database and shows it to you.