Partiality is the habit of favoring something — taking its part. If your parents always seem to let your little sister off the hook while you get grounded, you could accuse them of partiality in their parenting. They favor your sister over you.
Partiality is like bias. It means that your favorite person, type of food, or way of operating tends to get priority when you’re making up your mind. A synonym for partiality is favoritism. For this reason, anyone who’s not directly benefiting from your partiality will not appreciate it. It’s unfair!