What is PHPSESSID?

PHP uses one of two methods to keep track of sessions. If cookies are enabled, like in your case, it uses them.

If cookies are disabled, it uses the URL. Although this can be done securely, it's harder and it often, well, isn't. See, e.g., session fixation.

Search for it, you will get lots of SEO advice. The conventional wisdom is that you should use the cookies, but php will keep track of the session either way.

PHPSESSID reveals you are using PHP. If you don't want this you can easily change the name using the session.name in your php.ini file or using the session_name() function.

It's the identifier for your current session in PHP. If you delete it, you won't be able to access/make use of session variables. I'd suggest you keep it.