So, you've got, let's say, an air elemental you want to kill, and it's basically made of air. How do you kill it? Simply by hitting it with any weapon seems a little bit illogical to me - it's air and you can't kill it with a nonmagical spork... Or can you?
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An air elemental is not just air. It is a corporeal creature type and as such it has a "body" that can be disrupted by normal weapons.
Elemental traits from the back of the monster manual include only "immunity to poison, sleep effects, paralysis, and stunning" and "Not subject to critical hits or flanking", so its perceptions and body are not like a human's, with vulnerable points such as blood vessels, joints, or a brain, which would allow critical hits. However it does have a body.
Perhaps its body is made of air which has been bound together in some manner, but then again our bodies are in large measure just oxygen, carbon, and hydrogen bound together in some manner (that is, water and carbon compounds). Disrupt the organization of a body and you kill the creature, regardless of whether it is humanoid, dragon, elemental, or something else. Even incorporeal creatures can be disrupted by weapons, though only magically enhanced ones, to the point of destroying them.