The Twili are a race of beings who inhabit the Twilight Realm. Most of the inhabitants were turned into "monsters" known as Shadow Beasts or shadow messengers, under Zant's evil rule. Unlike the other Twili as shown in the game, Midna and Zant are the only Twili who retain their true appearance, although Midna was stuck in an impish form for much of the game. All other Twili that are briefly seen in Twilight Princess were quite different in appearance. This is a topic that has been debated among fans ever since; though the Twili look different compared to Midna and Zant, Midna mentioned herself that the Twili had been changed back to their original form.
Eons ago, the Twili were a group of dark sorcerers of unknown ethnicity referred to as Dark Interlopers, who were condemned to the Twilight Realm by the Golden Goddesses of Hyrule as punishment for their misgoverned attempt to use magic to claim the Triforce. Their "old magic" became known as the Fused Shadows, a powerful instrument. However, during their failed attempt to conquer the Sacred Realm, their Fused Shadow was split into four pieces and scattered throughout the two realms, The Twilight realm and Hyrule. The tribe lived and evolved in the Twilight Realm for centuries in isolation, mere shadows of what they once were, as over time they grew used to their surroundings and came to accept their fate. While some learned to love the beautiful peace that can be found only in Twilight, it drove other inhabitants mad.
After centuries of rule, Princess Midna became just one of many rulers in a long line of royalty, the Twilight Princess. It was around this time that a rebel force grew from within the Twilight, spearheaded by Zant. It is revealed that Zant believed that he was next to become Twilight Ruler, but the position was passed over. In a terrible fit of rage, he used the power of his false-god Ganondorf to transform most of the inhabitants of Twilight into monsters, and as a form of humiliation, transformed Midna into an imp creature. Midna's only hope was to find the magic of her ancestors, the Fused Shadows, and use it against Zant to save the realm and her people.
The Twili are usually seen as enemies, but when Link goes to the Palace of Twilight, he reverts several of the Twili to their true forms.
A Twili cleansed of Zant's curse by the light of the Sols
The Twili have strange glowing markings on their skin, which is pale bluish-gray in color partly covered by black. This complexion may be due to lack of full light as the Twili have changed from what they were when they were ejected from the "World of Light". They also have oval eyes that are normally red, yellow, or orange. They are (usually) quite thin as well as much taller than Hylians. Midna stood no less than than a full head above Link.
Midna's true form seems more human-like than this; she has distinct eyes and has many other characteristics that differ from the other Twili. This could be due to her lineage as there may be a separate sub-species of Twili in their Royal Family that have more human-like qualities in them, or at least females look more human-like, since Midna is the only known female Twili in Twilight Princess. It is also quite possible that once Ganondorf is killed, and Zant's spirit dies, that the Twili race as a whole are turned back into beings more resembling Midna and Zant.
A theory has been created that the dark patches on the Twili's body are not pigmentation as they appear to be, but what the Twili wear for clothes: shadows. Because fabric would probably be scarce in the Twilight Realm, Midna and Zant, being royalty, are the only Twili shown to wear fabric. Midna wears a simple cape and skirt while the rest of her body is covered in dark patches; Zant due to his vain nature wears a complete outfit covering up his entire body. All other Twili wear no fabric and have dark patches of skin in the shape of a toga with a hood.
Timeline, Ocarina and Majora's Mask
The events described in Twilight Princess are likely to have occurred before the events of Ocarina of Time, which is generally believed to be near the beginning of the timeline in Zelda game. This is because the Twili's ancestors' are described as trying to assert dominion over the Sacred Realm before the Triforce had ever been successfully found, and as the Triforce has been split by the time of Twilight Princess, claimed as being several decades after Ocarina of Time, it is likely that it was found during Ocarina of Time. Furthermore, the Twili's ancestors are described as having been imprisoned long ago, long enough for their race to lose all passion and ambition.
Though it has been theorized, Twilight Princess makes no mention of Majora's Mask. Debates arose when similarity between the eye-design of Midna and Majora's Mask were alleged by a number of fans, and now some players choose to speculate that the Twili are in fact responsible for creating Majora in a separate attempt to obtain the Triforce. However, this is contradictory, as the Twili were imprisoned by the Light Spirits for abusing their magic, while the ancient tribe that used Majora's Mask chose to seal it themselves.
Yet, others believe that they are the descendants of the Gerudo, although this is debatably unlikely due to the fact there are (or at least what appear to be) numerous male Twili. One would have to assume that they evolved to have more males at a time. However, evidence in favor of this idea is that Zant appears to have the Gerudo Symbol on his clothing (but this may be because he was currently a servant of Ganondorf, who is a member of the Gerudo race). In addition, the Twili have red hair, a common trait of the Gerudo.
They could also be descendants of the Sheikah, a question raised in part by the appearance of the cryptic Eye Symbol on the Fused Shadow - although it lacks the signature teardrop that the Sheikah tend to use. Midna's red eyes also cause suspicion here, but this detail may be simply coincidental (as is likely with Kafei too).
Finally, there remains the possibility that the Interlopers were not of any one race or tribe, but were a cult of people made up of several different races and cultures. Evidence to support this is Lanayru's vision where Link kills his closest friend to seek after the Triforce, only to become an Interloper himself. It can be interpreted that several people throughout Hyrule betrayed their fellow countrymen to obtain the Triforce and eventually united as the Interlopers. Once sealed in the Twilight Realm, the long-term effect of the Twilight on them could have transformed them into one single race, and may explain the large differences in appearance that some Twili have from others.
Veran and Onox
There are many theories that suggest Veran is actually a Twili. She has red hair, which is a common characteristic among the Twili, she dresses similarly to the Twili, and she also bears a striking resemblance to Midna's true form. She also bears what appears to be the Eye Symbol of the Sheikah, which many other Twili seem to bear. If Veran is a Twili, this would lend support to the theory of the Twili being Sheikah, though Veran's connection to the Sheikah is no less speculative than the Twili's, thus leaving room for doubt of the idea.
Onox also shares some qualities with the Twili, particularly Zant, both of them having very similar armor and bearing the Gerudo Symbol. However, the latter trait could merely be due to their services to Ganon and/or, in Onox's case, Twinrova. There is very little linking the two otherwise.
It is interesting to note that both of these antagonists are mentioned in connection with a "Dark Realm", which could coincide with the Twilight Realm. However, Onox's true form is the Dark Dragon, and while little is admittedly known about the native fauna of the Twilight Realm, no dragons are seen or even mentioned in the Twilight Realm when it is entered in Twilight Princess, thus making the idea that the realm has wild dragons speculative at best. Moreover, a location of the same name is briefly explored in Spirit Tracks, thus introducing the idea that this was in fact the two's place of origin, or at the very least that it is another realm unto itself.
Vaati bears resemblance to the Twili. It is also mentioned that Vaati comes from a darker place than the rest of the Minish.  It could be possible that he could have come from the Twilight. It could be possible that Vaati is a Twilit Minish.
While the Twili and their related artifacts tend to have structured, inorganic patterns, the Spirits of Light and Light objects are covered in curved and flowing markings. This may serve as further distinction between the two opposing sides in Twilight Princess.
↑"It was a peaceful place...until Zant took control of the Twilight Realm and transformed all of the Twili into shadow beasts." — Midna (Twilight Princess)
↑"Among those living in the light, interlopers who excelled at magic appeared. Wielding powerful sorcery, they tried to establish dominion over the Sacred Realm." — Lanayru (Twilight Princess)
↑"What do you think happened to the magic wielders who tried to rule the Sacred Realm? They were banished. They were chased across the sacred lands of Hyrule and driven into another realm by the goddesses." — Midna (Twilight Princess)
↑"Wielding powerful sorcery, they tried to establish dominion over the Sacred Realm. It was then that the goddesses ordered us three light spirits to intervene. We sealed away the great magic those individuals had mastered. You know this magic… It is the dark power you seek…the Fused Shadow." — Lanayru (Twilight Princess)
↑"In the shadows we regressed, so much so that we soon knew neither anger nor hatred...nor even the faintest bloom of desire." — Zant (Twilight Princess)
↑"Because I believed I would be the next to rule our people! THAT is why! But would they acknowledge me as their king? No!" — Zant (Twilight Princess)