Family Boss Fights pit your entire family against an epic boss. In the case of Jeff The Knife, you'll have 7 days to beat him. (You may have more or less time to take out future Family Bosses.) Whenever the timer runs out on a Family Boss Fight, the boss' health resets and you will have another chance to prove your family is in charge.
Step 1: Choose A Role
When you first join a Family Boss Fight, you will elect to participate in one of three roles, each possessing a different special ability. Bruisers enable fellow family members to deal double damage, Arsonists reduce a Family Boss' rage, which would otherwise prevent attacks from dealing full damage, and Racketeers reduce the fatigue of fellow family members, helping them to continue to fight. You will have the role you choose for the duration of the Family Boss Fight and will only be able to choose a new one when you decide to participate in another round after a timer reset.
Step 2: Attack!
There are two ways to inflict damage upon a Family Boss: 1) Use stamina to deal an amount of damage determined by your attack skill and the Family Boss' current level of rage, and 2) Use Family Boss Fight ammos.
Step 3: Rally to Win
Request role-specific buffs from your family to help defeat the Family Boss. As explained earlier, participating family members select a role when they enter the fight, each of which can send a different type of buff to other family members (in other words, you can't use a buff on yourself). You can ask for a buff via the Buff Request Bar across the top of the Family Boss Fight window. When a family member is in need of the type of buff you can provide, you'll see a Buff Request icon appear over their score in the leaderboard. Clicking that icon sends a buff to them. You begin with 5 of the buffs of your chosen role, after which you'll have to wait for them to regenerate at a rate of 1 every 30 minutes.
Family Member Leaderboard & Loot
No matter your role, your dedication can earn you a spot at the top of the leaderboard, which shows how much each of your participating family members are contributing in the fight. And there's good reason to try to be at the top, as the awesomeness of the loot you will receive if your family defeats a Family Boss depends on the degree to which you participate in the fight. Put in an OK amount of effort and the loot you could receive will range from OK to Great, but, do more to help in the fight and your loot will range from Great to Epic! On top of that, you can collect the 4 top loot items to land an achievement!
The Boss Rage Bar
As alluded to earlier, a Family Boss has a rage bar, which fills up a little bit more each time the they are attacked. As a Family Boss' rage grows, attacks become less effective and the amount of damage they receive is reduced. Be sure to ask your family Arsonists to cool the boss down so your attacks will inflict the maximum damage possible. Also, ask your family Bruisers to 2x your attack and show the Family Boss who's really in charge!
The Fatigue Bar
Mobsters in every role can attack a Family Boss, but it costs stamina to do so. Exactly how much stamina your next attack will cost depends on your level of fatigue. The more fatigued you are, the more stamina it costs to attack, and therefore the fewer attacks you can make with the stamina you have, so be sure to ask your family Racketeers to reduce your fatigue to keep you fighting longer. And then ask your family Bruisers to 2x your attack so they count that much more!

–Mafia Wars Blog