Bonds are a way to give character history with each other common to PBtA games. They are a chance for players to talk about specific moments in their past that they might have shared with others.
For example in Nightwitches this is called Regard and takes the form:
I (love/trust/admire/respect/hate/resent/pity/fear/etc) (name of person or aircraft) the most.
In Pigsmoke they are question you have to answer about your character, often to create NPC attachments, for example:
Who has roped you into most of the work on their latest project? What is it? Why are you helping?
Define external redirect: Flags