Loosely, affinity determines a character's stat boosts that are gained from supports. Supports depend on the affinities of the two characters that are being bonded, and they determine the type of bonus given. The following tables are a compilation of bonuses given by the fire affinity in each game.
The Fire affinity gives offensive combat bonuses, and is usually desirable for characters with low attack power, or to maximize offensive potential with stronger characters. In the GBA games, it is considered superior to the similarly offensive Light affinity, because it gives an evasion bonus as well. Along with the Light affinity, it gives no Critical Evade on its own.