If my character is attacked by a Flame Tongue Maul for example, which does an additional 2d6 fire damage on top of its 2d6 bludgeoning, and then I use my reaction to cast the Absorb Elements spell, do I gain resistance to bludgeoning damage, or fire damage?
Take note of the casting time of this spell, particularly the 'condition' required to be able to use it:
1 Reaction, which you take when you take acid, cold, fire, lightning, or thunder damage
Then the text saying
The spell captures some of the incoming energy, lessening its effect on you and storing it for your next melee attack. You have resistance to the triggering damage type until the start of your next turn. Also, the first time you hit with a melee attack on your next turn, the target takes an extra 1d6 damage of the triggering type, and the spell ends.
So, you can only use Absorb Elements when you take acid, cold, fire, lightning, or thunder damage. Whatever damage type allowed you to use the spell is the "triggering damage type," (the type of damage that allowed you to trigger the spell) so that's the damage type you get Resistance to.
Bludgeoning damage is not on the list of damage types that can trigger Absorb Elements, thus it can't give you any protection against it.