A bonding event (絆イベント) is an event-based system to bond with a selection of other characters, first introduced in Zero and Ao no Kiseki and further refined throughout Trails of Cold Steel I, II, III, and IV.
Zero and Ao no Kiseki
Trails of Cold Steel I, II, III, and IV
On free days, special events can be triggered with students of Class VII and certain other characters. One point will be consumed each time you view an event, but your bond with that character will be strengthened. On the evening of each free day, you will receive one bonus point, which can be used to trigger one evening-exclusive bonding event.
The characters with bonding events are highlighted by the Yellow Exclamation mark, shown on the Trista Map shortcut. Speak to either of them by choice to gain Link EXP with them, while simultaneously completing the character notes in the Student Notebook.