It's a mix. DR is described as varying from stuff like instantaneous regeneration (you wound it, but the flesh heals immediately) to sheer immunity (your weapon hits but fails to dig in or pierce the flesh, bouncing off harmlessly).
I figure it's all up to DM fiat and what seems most appropriate to the creature.
Something with DR X/adamantine? Obviously it's really frikkin' tough, so non-adamantine stuff just bounces off as surely as if you were trying to hack away at an adamantine wall with an ordinary steel blade.
Something with DR X/magic? Could be instant healing or sheer immunity, or something else. Maybe the stricken spot turns to vapor for a moment, until your weapon passes through, then resolidifies without any apparent harm done?
Something with DR X/silver? Probably instant healing; your steel-tipped arrow digs in, then pops out a second later, as the flesh rapidly knits itself and recovering muscle fibers push out the arrow.
Something with DR X/cold iron? Likewise instant healing, probably; your hammer strikes the cruel fey, but bones snap back into place and mend themselves in seconds, the flesh recovering its normal shape and vitality almost as quickly as the wound was dealt, the fey creature sneering at your unnatural, technologically-forged warhammer with only the briefest feeling of pain and inconvenience.
Something with DR X/alignment? Probably sheer immunity/repulsion; your spiked chain barely even touches the fiend's flesh, leaving no marks, only loosely wrapped around its arm, easily shaken off with a snap of the arm, and your buddy's mace just rebounds off the fiend's profane flesh as though it were a sturdy wall of adamantine, repelled from making any repugnant contact with the flesh of the damned.