Sessions are triggered when an enemy's weakness is attacked by a skill and if one or more of your allies have the corresponding Session Skill to perform a follow up attack. To know which characters can perform session attacks, you have to refer to their Skill Menu. Session Skills are inherited from various Carnages so make sure to inherit all skills and create new Carnage Unities whenever you can.
During the battle, you'll know which character has a session with your specific attack since their backdrop portrait will change as well.
In the example image above, Tsubasa has two session skills: Elec-Lunge and Sword-Lunge. This means if anyone in the party uses an Elec attack like Zio or a Sword attack like Cleave, she'll perform a Lunge follow up-attack. Take note the the session skill's attack type will still be considered against the target's resistances. Using the same example above, if the target is resistant against Thrust-type attacks, Tsubasa's follow up attack will deal lesser damage even if it's within a session.
During a session, if the current target is defeated, another target will be selected automatically and the remaining attacks of that session will ignore the next target's Resist, Null, Repel, or Drain. Not only that, the third attack in a Session usually "steals" an item or money from the target.
These flashy yet powerful special attacks can be triggered once you have filled the SP Gauge in the upper right corner of the screen. Special Performances can trigger Sessions while ignoring enemy affinities and usually have greater damage output and beneficial effects to the party compared to other skills. The gauge is filled slowly by performing actions and chaining attacks through successful Session skills.