A bond is a type of trait that describes a personal connection between a player character and one or more other characters, primarily other player characters. A bond is generally a short phrase or sentence that describes how one character feels about another character. Bonds between characters might be symmetrical (both characters have the same bond on their own sheet) or asymmetrical (a character's bond for another character could be very different than that character's bond for them). It is also possible for a character to have multiple bonds with the same character, which may complement each other or contradict each other (introducing complexity into the characters' relationship).
Bonds are often used by game systems to either encourage cohesion in a party or to introduce intra-party drama or conflict, depending on the goals of the system in question.
Some game systems use bonds that have a mechanical component. For example, a bond might have an associated score or rating that interacts with other game mechanics (the game Smallville has bonds attached to rated relationship traits that can only be used when the bond is relevant) or bonds might be an element in the game's reward cycle (bonds in Dungeon World provide xp when they are resolved). Other bonds may have no mechanical component and instead provide narrative hooks for players, like character relationships in Fiasco. Bonds in Dungeons & Dragons 5th edition are primarily narrative hooks, but the game also encourages the GM (called the DM in D&D) to give players inspiration (one free use of advantage on a future roll) if they roleplay in a way that's true to their bond.