You can Declare War if you see this banner...
...or by clicking on Fight menu and clicking the section with the yellow star below:
Once you've declared war - or if you see that war has been declared against you, rally your Mafia to help you, with a call for help on your news feed:
You need 7 of your Mafia members to win a fight against your opponent to win the war.
Each time you win a war, you will obtain TWO of the 16 War Rewards - loot items of increasing value (you might get two of the same Reward thought). The final four of these are better than anything in Vegas District 8.
Once you obtain all 16 of the War Rewards, you will also earn the Warlord Achievement (and a Skill Point).
Another nice addition to the game in what seems like an unprecedented week of Mafia Wars developments...