Demons have at least three different levels, which you can tell by their eye color. Lowest-level demons have black eyes; higher level have yellow; the highest have white. (However, the crossroads demon has red eyes: she seems to have answered to Azazel at some point, but now answers to Lilith. Possibly.)
They leave sulfur behind, and possibly also EMF energy.
They can possess any human, including children. Lilith’s favorite is little girls. The possessed person is usually aware of what their body is doing. A demon will also keep a body going after it’s too injured to function on its own, and if a body is broken and the demon is driven out, the person will die.
Outside of a human body, demons appear as thick black smoke.
They are lawful evil, basically: governed by strict laws and rules. They can be destroyed (sent back to hell?) by Ruby’s knife or the Colt, or driven out of a body by exorcism rituals. They cannot step outside of a devil’s trap symbol, and cannot possess a body with a protective artifact or symbol on it.
In Supernatural, the low-level demons are human souls that have been twisted by Hell’s torments, while the higher up demons like Azazel, Lilith, and Alistair are pure, ancient demons, and we don’t know much more about them than that.
SPN demonic omens:
*lightning storms
*cattle deaths
*temperature fluctuations
Azazel is mentioned in Leviticus: scapegoats, the personification of wickedness, were sacrificed to him. Symbol of impurity. Might have been a fallen angel. In the Book of Enoch he’s one of the Grigori who fell, and is a bringer of forbidden knowledge.