The Dodge action is primarily used to avoid attacks. It causes the attacker to have disadvantage (roll twice, take the lowest). The dodging character doesn't make any additional rolls against attacks.
Dodge also grants the target advantage (roll twice, take the highest) on Dexterity saving throws.
Both benefits require the dodging character to see the incoming attack.
Saving Throws are more frequently the result of spellcasting or secondary effects of creature attacks, rather than traps. Some spells use an attack roll made by the caster, while others cause a save by the target. They are used in a context where their effectiveness is not based on effort or skill of the attacker but by failure of the defender.
Some creatures (e.g. Giant Spider) inflict poison with their attacks. While their attack roll determines if the actual attack makes contact, it has no bearing on the effectiveness of the venom. If the conditions are satisfied (target is dodging and can see the attacker), they'd make this attack roll with disadvantage.
If the attack hits, the target makes a Constitution saving throw (aiming to beat the save DC described in the attack) to determine if their body is capable of resisting the effects of the venom. Dodge has no effect here, the attack has already landed.
By contrast, a spell like Fireball requires a Dexterity saving throw to reduce damage. A Dodging character would receive advantage on this save, as long as he can see it coming.