After choosing a Legacy surname (which many players have already done), BioWare will allow you to connect your characters through a family tree in the Legacy panel. Characters don't have to just be blood relatives; they can be allies, rivals, and adopted family members as well.
Wallace explained that through these Legacy connections, characters have inherited many traits that run strong in families. These unlocks come in the form of shared emotes (which come at the end of Chapter 2), shared heroic abilities (unlocked at the end of Chapter 3), and the ability to roll the same race no matter what the class (unlocked at level 50). The Legacy system will also allow companions to unlock passive stat bonuses shared across all companions of that same archetype. Ship improvements, including mail, GTN access, and droids, are other perks of this system.
However, players will have to wait until Update 1.3 to purchase per-character perks. These, as Wallace hints, will allow players to "customize the way their characters progress, as well as new rewards, conveniences, and all kinds of other fun surprises."