How a site remembers you
When you sign in, a website may start a session and give your browser a cookie with an ID for it. Your browser sends the cookie with later requests, so the website can recognise you as you move around the site.
Often, the website keeps the session record on its server. Some sites keep session information in the cookie itself. A stolen sign-in cookie could let someone use your session, so it needs protection.
More than sign-ins
Cookies can also remember a basket or preferences, or help with analytics and advertising. These uses have different privacy implications. Websites can restrict when cookies are sent and prevent page code from reading sensitive ones.