An Aoandon?
Yeah, that's a pretty confusing attack. (Confusing to me, not just to PCs who fail the Will Save.) It does 10d6 damage. Not any particular type of damage, just damage in general.
As untyped damage, it probably pierces all DR and resistances, similar to the Lantern Archon's attack
It doesn't do normal weapon damage so DR won't help.
DR applies only against an actual weapon (both manufactured and natural). If an incorporeal attack is coming from some special ability, like a ghost's corrupting touch power, it bypasses DR unless the ability specifically says otherwise.
An actual weapon is not necessary. Damage Reduction can apply even against magic attacks. It is damage type that is looked at. Anything that deals bludgeoning, piercing, or slashing damage is subject to damage reduction, regardless of its origin.
If a magic spell, special ability, or touch attack deals at least one of those types of damage, damage reduction applies. Granted, I don't know of any incorporeal touch attack that does that kind of damage, but the possibility remains.
(And before someone brings up "But the Bestiary says...":)
Damage Reduction: How does DR interact with magical effects that deal bludgeoning, piercing, or slashing damage?
Although the Bestiary definition of Damage Reduction (page 299) says "The creature takes normal damage from energy attacks (even nonmagical ones), spells, spell-like abilities, and supernatural abilities," that's actually just referring to damage that isn't specifically called out as being of a particular type, such as fire damage or piercing damage. In other words, DR doesn't protect against "typeless damage" from magical attacks.
However, if a magical attack specifically mentions that it deals bludgeoning, piercing, or slashing damage, DR affects that damage normally, as if it were from a physical weapon. (Otherwise the magical attack might as well not have a damage type, as it would only interface with B/P/S damage in a very few corner cases, such as whether or not an ooze splits from that attack.)
For example, the ice storm spell deals 3d6 points of bludgeoning damage and 2d6 points of cold damage. If you cast ice storm at a group of zombies, the zombie's DR 5/slashing protects them against 5 points of the spell's bludgeoning damage. Their DR doesn't help them against the spell's cold damage because DR doesn't apply to energy attacks.
Damage reduction does not negate touch attacks, energy damage dealt along with an attack, or energy drains.
Pretty clear the aoandon's Touch of Madness is unaffected by DR.
/cevah