Score

Score reflects how capable your Character is at competing with all the other characters you are fighting. The score of your opponents is a primary factor in your Challenge Bonus.

Here are some details about how score is calculated:

Most obviously, you will gain more score for beating higher-ranked players, and lose less for losing to them. Conversely, you will lose more score the farther below you your opponent is.
The more battles it takes you to beat someone, the less his score will decrease from the loss. This does not affect how much score YOU get from the win, or how much money and Experience (XP) you get. It is worth noting that you might very well not be able to lower a higher-ranked player's score in multiple battles because he is earning as much from the wins as he loses from the loss.
If the winner is ranked below the loser, their scores will converge fairly rapidly. Otherwise, there is a minimum amount of score gain/loss that will apply. The score penalty for losing tends to be greater when you are the challenger.
The system prefers to keep your score near your Power. If you win against someone while your score is above your Power, you will gain less score than if you had beaten that person while your score was less than your Power. This only compares YOUR score with YOUR Power, not to your opponent's. So someone whose Power is above yours can still lower your score.
The loser will lose score and the winner gain even if the winner's score is already near or above the losers. The amount of this "alternative minimum trauma" depends on each opponents score.
The only score updated in real-time, so to speak, is the one for your active player on your home page. Use the Update feature on the Inspect page to make sure you are getting accurate stats on a Character.
Some confuse Score with Rank