A cookie is basically just an item in a dictionary. Each item has a key and a value. For authentication, the key could be something like 'username' and the value would be the username. Each time you make a request to a website, your browser will include the cookies in the request, and the host server will check the cookies. So authentication can be done automatically like that.
To set a cookie, you just have to add it to the response the server sends back after requests. The browser will then add the cookie upon receiving the response.
There are different options you can configure for the cookie server side, like expiration times or encryption. An encrypted cookie is often referred to as a signed cookie. Basically the server encrypts the key and value in the dictionary item, so only the server can make use of the information. So then cookie would be secure.
A browser will save the cookies set by the server. In the HTTP header of every request the browser makes to that server, it will add the cookies. It will only add cookies for the domains that set them. Example.com can set a cookie and also add options in the HTTP header for the browsers to send the cookie back to subdomains, like sub.example.com. It would be unacceptable for a browser to ever sends cookies to a different domain.