Naturally you must be logged into your GMail account in order to be able to delete it. If you can't login you can't delete the account; it's that simple. This prevents just anyone from deleting your account.
"... there's no 'undo': once you delete your email account it's gone forever and cannot be restored."
Once you're there, go to "Accounts and Import":
Part way down the resulting page in the section labeled Google Account settings, click on the link labeled Google Account settings: Read what they say:
That's the default setup: Gmail, iGoogle and Web Search history. click on "Edit" next to My Products... at the top of that list:
Now you can finally see what's going on! The question is: do you want to delete your Gmail account, or do you want to delete the entire account and all Google services?
I'm going to assume the latter, though if I'm wrong, you can see what to select here. I click on "Close account and delete all services" and get this long, complicated survey:
Ya really want to proceed? Check all the boxes, enter your password again and click on "Delete Google Account", and...
Done....