I become a fan during the '01-02 season when I was about 13. Of course, my house was always a Celtics house so I knew a fair amount about the team before that but I remember the Celtics chasing the Nets for the Atlantic title and getting into the playoffs big time.
'02-03 was, personally, a favorite of mine because after the Conference Finals appearance I really threw myself into the team, thinking it was nothing but blue skies ahead for us. Of course, the next bunch of years were a series of reality checks for me. So that made the '07-08 season pretty sweet when it all came together.
If I remember, the '06-07 season wasn't so bad because there was a sense of building toward something (with Big Al and Tony Allen and Delonte West and Gerald Green, who was so much fun to watch) but that '05-06 was wretched.