As stated all over the place, the goal of the game is to become the new head of the Hearthstone family, by gaining power and influence. But what exactly does this mean, and how is it accomplished?
First of all, it should be noted that there are a number of different endings possible in the game, and the one which befalls your character is determined by what you have done in the game. If you've made a habit of teasing Lucifer throughout the game, and he ends up patriarch, your ending's probably not going to be very pleasant; whereas if you're his staunchest friend, you'll have a great ending.
Also, there are endings which do not tie directly into the main plot. If you're Betty and you become the Queen of the Underworld, or something, you may not be head of the Hearthstone family but you've certainly "won", in your own way.
All the other siblings feel the same way; there are things they value more than heading the family, such as security and happiness. If you have both the peasantry and gentry behind you, the fact that Arthur could wipe them all away with his Super-Apocalypse-Ultimate spell doesn't really matter, because his doing so would jeopardize his way of life, and greatly reduce the power of the family, something he wouldn't do. Essentially, it's better to be a normal member of a powerful Hearthstone family than the head of a weak one. And a large part of the Hearthstone's power is the continued support of the surrounding areas, though they're loathe to admit it.
This is why influence matters: if others support you, your siblings will have a more difficult time challenging you, because doing so will bring on the enmity of the community. However, if the parties backing two contestants are relatively equal, an altercation is bound to arise, and this is why power and big, scary friends matters.
I'm not sure, yet, how exactly the mathematics of determining the victor will work, as there are just so many factors to work out. I do know that, if all else fails, you will be allowed to make open war on your siblings to take control, so if you've been spending all your time on "push-ups and crunches" quests, this is your moment to shine.