The siren is one of the more dangerous supernatural creatures the Winchesters face during their journey. Being able to shapeshift and to quickly find out what to do to appeal to their victims, before enthralling and forcing them to commit heinous crimes against their loved ones, most likely killing them in the process. Not even the winchesters were able to escape their claws and one can only speculate what would have happened if they weren’t saved by Bobby.
In greek mythology the sirens were only three women in the shape of bird-women, luring nearby sailors in with their enchanting voice, causing them to shipwreck on the shores of their rocky cliffs, killing them in the process. There are sirens in many local lores, all share the same bird-like appearance to them, as well as the ability to lure men especially to their shores to kill them afterwards. Later their bodies were changed, so not only their voices (like those of birds) are pretty, but their bodies too so it would easier for them to seduce their victims. Only the ability to be able to lure people in remained in the version of Sirens in Supernatural, though they are able to shapeshift into something their victim desires. They are able to read the minds of their target to find out what they desire. Like this the siren Sam and Dean faced in season 4 was able to become what Dean wanted: A person who shares the same interests with Dean and who was able to replace Sam, to whom Dean's trust suffered particularly badly this season due to his shenanigans with Ruby and his addiction to demon blood, which made him become a demon more and more, a fact that frightened Dean.
To compulse somebody, they use the venom glands in their mouth. The venom generates a high dose of oxytocin in the body of their victims, giving them a feeling of intense euphoria and deep love for the siren. While in this state, they are entirely devoted to their loved one, ready to carry out drastic actions just to make their siren happy and to gain their affection. The siren gains euphoria themselves once a serious crime like murder was committed in their name, usually getting bored shortly after, which makes them seek a new victim to play with and rinse and repeat these actions on.
Their biggest weakness is a bronze dagger coated in the blood of one of their victims. They also can be killed when they receive severe trauma to the head, as shown when Dean beats a siren to death with a baseball bat. They are not immortal, but pretty much invulnerable through every other means of harm, unlike their mythological counterparts, which turn to a part of the cliff if they’re unable to kill the sailors they’re singing to.
A mirror also shows their real face, which makes speculation possible that sirens are not shapeshifters, but actually just create illusions to deceive their victims.
Sirens are often equated with mermaids (for example in Pirates of the Carribean: On Stranger Tides), but in mythological lore mermaids are almost always perceived as friendly and harmless merpeople, while sirens are always depicted as harmful and dangerous creatures.
They are also often described and depicted as harpyies (for example in the video game Baldurs Gate 3, where they look like sirens and use their alluring voice to capture their prey).