The session data that you read and write using $_SESSION is stored on server side, usually in text files in a temporary directory. They can not be accessed from outside.
The thing connecting a session to a client browser is the session ID, which is usually stored in a cookie (see the comments for exceptions to that rule). This ID is, and should be, the only thing about your session that is stored on client side.
If you delete the cookie in the browser, the connection to that session is lost, even if the file on the server continues to exist for some time.
The session.save_path variable influences the location on the server where the session data is stored. If you are not the server’s administrator, it is usually not necessary to change it.