I would like to have query that would select from the database all the information which is private = 0 unless the user = 'username'
SELECT * FROM table WHERE private = 0 "unless" user = 'username'
Answer to original question:
SELECT * FROM table WHERE private = 0 and user <> 'username'
EDIT: OP added the following comment:
What if I want to choose where private = 0 unless private = 1 and user = 'username'
Then it sounds like you simply want this (but it's not 100% clear):
SELECT * FROM table WHERE private = 0 OR (private = 1 and user = 'username')