The attack skill has two (2) main uses. When you initiate an attack (whether through Fight, Rob, or attempting a Hit), your total attack value is compared to your defender's total defense value. Your attack value also has a minor impact on the amount of damage you cause whether attacking or defending.

Fighting and Robbing gets into more details over how the fight rolls are actually made, but here is a quick summary. For your mafia size, X, the game chooses your X best items with the highest attack value, X best armors, and X best vehicles. Outside of special loot items and before level 100, this means a Chain Gun at 16/14, a Body Armor at 4/7, and a Town Car at 10/12.

Roughly every member of your family (assuming you have enough items for all of them) will add 30 attack and 33 defense. So 6 levels worth of skill points in attack is the equivalent of bumping your mafia size by 1 (whether through real people or the Godfather hired guns). Once your level is above 150 and a lot of your opponents will have a mafia size of 501, attack points can be useful. At low levels, it is a complete waste and dwarfed by your mafia size.