Well, technically when you log into a website (e.g. BA), the site sends you a cookie that you store on your browser. Every time you access the BA site your browser voluntarily sends that cookie.
BA has a copy of the cookie and can essentially use it to compare and authenticate you. The cookie has an expiration date after which you might be automatically logged out of the BA site. Depending on how they implement it they might not delete the cookie on your side but use it to say "we are sure it's petez but it's not enough to give him access to his account".

If you don't want the second part to happen then you would need to delete your cookies or not accept them. Your browsing experience might suffer, though.
This alone isn't enough to fully anonymize your browsing experience. Through unique information (browser version, operating system, installed plug-ins, screen resolution, installed fonts, ...) they might still be able to identify you.