As you do jobs and fight, you will have a random chance for this pop-up to appear:
Apparently, players can enter this secret code (blurred for privacy, of course!) in at the Mafia Wars app home page for a special bonus each day.
Many players have not reported seeing a place on the home page to enter the code. There seems to be a refractory period (of about a few hours) between which you get your code and the space appears on the homepage that allows you to edit it. The box to enter the code is big, so once it is available you can't miss it:
Additionally, this bonus is also only meant for one account - if you enter the wrong code into a different account, you won't get a bonus (or worse, who knows), so make sure you keep your accounts straight and do not share your code with friends.