Firstly, you can send mail from any address (even fake) which you like, for example, email@example.com
or even a word biledeamon in the sender (from:) field. However, don't expect that your email recipient will know how to respond, because none of these emails are real (matches real your address). To give yourself firstname.lastname@example.org
, you at first must buy this domain or have a good friend who already owns it. If you don't have money, you can get a free second level domain (yourname.somedomain.tld) with dyndns.org or afraid.org. Now your mail adress would be email@example.com
(same as you have said).
E-mail uses the same principle as real post. You can write anything in from field and you must put correct address of recipient. Same vice versa.
I hope, I explained it correctly. You can read more about dns and it would be clearer for you.