Best option for Session management in Java
The session management (client identification, cookie handling, saving session scoped data and so on) is basically already done by the appserver itself. You don't need to worry about it at all. You can just set/get Java objects in the session by HttpSession#setAttribute() and #getAttribute(). Only thing what you really need to take care of is the URL rewriting for the case that the client doesn't support cookies. It will then append a jsessionid identifier to the URL. In the JSP you can use the JSTL's c:url for this. In the Servlet you can use HttpServletResponse#encodeURL() for this. This way the server can identify the client by reading the new request URL.
Your new question shall probably be "But how are cookies related to this? How does the server do it all?". Well, the answer is this: if the server receives a request from a client and the server side code (your code) is trying to get the HttpSession by HttpServletRequest#getSession() while there's no one created yet (first request in a fresh session), the server will create a new one itself. The server will generate a long, unique and hard-to-guess ID (the one which you can get by HttpSession#getId()) and set this ID as a value of the cookie with the name jsessionid. Under the hood the server uses HttpServletResponse#addCookie() for this. Finally the server will store all sessions in some kind of Map with the session ID as key and the HttpSession as value.
According to the HTTP cookie spec the client is required to send the same cookies back in the headers of the subsequent request. Under the hood the server will search for the jsessionid cookie by HttpServletRequest#getCookies() and determine its value. This way the server is able to obtain the associated HttpSession and give it back by every call on HttpServletRequest#getSession().
To the point: the only thing which is stored in the client side is the session ID (in flavor of a cookie) and the HttpSession object (including all of its attributes) is stored in the server side (in Java's memory). You don't need to worry about session management youself and you also don't need to worry about the security.
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All Java web frameworks support cookies or URL-encoded session IDs. They will chose the correct approach automatically, so there is nothing you need to do. Just request the session object from your container and it will handle the details.
[EDIT] There are two options: Cookies and a special URL. There are problems with both approaches. For example, if you encode the session in an URL, people can try to pass the session on (by putting the URL into a mail, for example). If you want to understand this, read a couple of articles about security and build app servers. Otherwise: Your Java app server will do the right thing for you. Don't think about it.
The cookie just stores the session ID, this ID is useless once the session has expired.