From a technical perspective, the definition of a Cookie can be found here. Loosely, think of a cookie as a piece of data returned by the Web Server you connected to in the past. This data is associated with the host that returned that data and can never be seen by other hosts. When you subsequently connect to the host in the future, the previously returned value (the cookie value) is sent back to the server. This allows the server to generate some data that can be used to "remember you" when you subsequently come back.
A session cookie is still "just a cookie" but is used to maintain "state of the session". For example, imagine a shopping cart. If you place items in your cart, the server will send back a cookie value that is a key used to find your cart again. If you place items in your cart today and come back tomorrow, the server can use the cookie value to lookup your cart.
As for "ending a session" ... this can be done at the browser by asking the browser to "forget" the cookie such that when you subsequently visit the web site, there is no cookie and hence it has no knowledge of your past interactions. Alternatively, the server can choose to ignore any cookie value you sent. A cookie can also have an implicit self deletion value such that if a time has passed, the cookie evaporates. Finally, the server can ask for the cookie value to be replaced / deleted when you next visit it.
I would suggest having a good google at Cookies in general as there are a lot of good references to how they work and how they are used.
