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Elves on the Discworld are based more on the nastier kind of fairy-folk in European (and other) folklores than elves as portrayed in most modern (post-Tolkien) fantasy fiction. They are not native to the Disc, but come from a "parasite universe", sometimes called Fairyland. This pocket dimension can latch onto different universes at certain times, allowing the elves (and other fae creatures such as unicorns, Jenny Greenteeth and, before they were cast out, the Nac Mac Feegle) to go hunting.
Elven invasions of the Discworld are recorded in Lords and Ladies and The Wee Free Men. The Science of Discworld II: The Globe involves an attempted invasion of Roundworld (a version of our world). In Moving Pictures it is mentioned that a few elves have moved to Holy Wood.
Though elves lack the ability to procreate within their own species, they are capable of breeding with humans, resulting in offspring with superficially elvish characteristics - skinny, pointy ears, a tendency to giggle and burn easily in the sun - but fundamentally human traits i.e. empathy. As mentioned above, creatures from the elves' universe often go hunting when they manage to enter a 'real' universe, one of the things they hunt being people. Elves have no proper imagination or real emotions, and therefore such things fascinate them. Because they cannot create they steal musicians and artists. Because they cannot have children they steal children from the Disc to be their toys. Because they cannot feel empathy they enjoy the suffering of others. In Lords and Ladies, Pratchett states that elves would smash the world if they thought it would make a pretty sound. Pratchett also compares the elves to cats, both being extremely stylish and excessively cruel.
Even if an elf is, for reasons of its own, trying to be nice, its lack of understanding of humans means that there is always something "off" about it. Mostly they get away with this, due to the illusion-creating glamour they cast. While elves are, as mentioned above, not musical, elfsong is perceived as beautiful by humans, and is highly hypnotic. When the Queen of Lancre, Magrat Garlick is released from its hypnotic effects, she realises that elven music really sounds like a saw being played. Elves are generally seen as innately beautiful and stylish, but this is just another aspect of the glamour. Some of them are only vaguely humanoid.
Elves have copper-based, blue-green blood, and are extremely vulnerable to iron. This is partly because there is no way of magicking iron, and partly that they have the ability to detect magnetic fields, and use them to navigate (as pigeons are thought to do) and even sense the thoughts of others from short distances. The presence of iron disrupts this and sends them into a panic. The main gateway to the world of Elves in Lancre is surrounded by several large boulders of naturally magnetised meteoric iron. When there is no iron around they have the absolute confidence of beings who know exactly where they are.
Elves are ruled by a King and Queen. They act much the same as the King and Queen in chess, the Queen is the more obvious ruler, organising the raiding parties and maintaining absolute control of her kingdom. The King tends not to get involved if he can avoid it and keeps to himself in a different aspect of Fairyland. The King and Queen may once have ruled together, but it has been implied there was an acrimonious split, which brings to mind the rift between the fairy courts of Oberon and Titania in A Midsummer Night's Dream.
If it comes to that, there may not only be one King and Queen. However, due to the elves' lack of imagination and creativity any Kings and Queens of the Elves tend to be similar.