No, you do not need to delete an object after its use in PHP. Let PHP's garbage collection take care of it for you.
There may be some obscure performance reasons to unset an object after you are done with it in very specific situations. But such cases are not everyday events, and they will be clear when you see them.
You're in the right neighborhood.
Most PHP applications will have active sessions for all page views, regardless of whether a user is logged in. It would essentially do a session start() at the beginning of the script, and a save() at the end. That session would exist for the entire visit to the site, or longer, depending on how you have PHP set to expire cookies.
When a user submits a login form and successfully logs in, you can then store the user information as a session variable. For example, at the most basic, you can store the user id associated with the session:
$_SESSION['user_id'] = 1234;
When the session is saved, that value is stored with the session. Session variables are not sent to the client - they are stored on the web server (where exactly it is stored is configurable, but ideally in a database).
Now, on the next page load, you can check the value of $_SESSION['user_id'] to get the user id of the logged in user, if any. This is because when the session loads at the top of the script, it has access to the session variables you assigned on other pages in that session.
When a user logs out, you set the $_SESSION['user_id'] back to 0. On the next page load, the user is no longer associated with that session.
If you want the user to get logged out when the browser is closed, you need to set the cookie lifetime to 0. If you want the session to persist, you want to set the cookie lifetime to the maximum number of seconds you want the session to exist (for example, 60 * 60 * 24 * 30 would be 30 days).
This is just a conceptual overview. There's some security issues involved with sessions, so make sure you learn about them as well before you make a site live.
Best Answer
For the purpose of safe coding, you should always close database connections explicitly to make sure that the code was able to close itself gracefully and to prevent any other objects from reusing the same connection after you are done with it.
Using unset is the same as leaving the database connection open, as you're relying on the garbage collector to clean up after the variable as opposed to asking the connection to gracefully close the connection after you're done with it. Though as mentioned by CodeCaster below, doing so is acceptable since the garbage collector makes sure that the connections are closed.
A useful example, lets say you're eating lunch at a cafeteria somewhere. After eating, you have two choice, throw your left overs in the trash then return the plate/tray to its rightful place(thats cleaning up after yourself). And the other is just leaving your mess on the table and wait for someone else(the garbage collector) to clean it up for you.
If you clean it up yourself, you're sure that your mess is cleaned up. But if you rely on the garbage collector, right before the garbage collector cleans up your mess, what if some weird stalker decides to use your used utensils to aid them in their nightly fantasies about you, then return them back to the cafeteria just in time for the garbage collector to clean up your mess. Stuff like that can happen.
EDIT: Corrected my mistake about garbage collectors.