If the Twilight Realm is the "anti-Hyrule", then the Twili are the anti-Hylians. This is the race to which Midna belongs.
History of the Twili
According to the light spirit Lanayru, when Hyrule was created the goddesses distributed power equally among its denizens. The place where they descended came to be known as the Sacred Realm, and was desired by people all over Hyrule for the power it contained; they fought over it, and amongst them appeared those who excelled at magic and wished to use this power to take over the magic.
When the light spirits intervened, they took this magic — the Fused Shadow — and sealed it away (although apparently they left a piece to the Twili's Royal Family, as Midna wears a piece as a crown). They banished those who had wielded it to a world of perpetual twilight, and those who inhabited it "became shadows that could not mingle with the light". The two worlds were joined by an artefact known as the Mirror of Twilight, which could only be activated from the world of light and was indestructable except by the true leader of the Twili.
Later, Ganondorf invaded Hyrule. He was the leader of a band of theives and a wielder of dark magic who wished to control the Sacred Realm. Because he was arrogant he was blind to failure and was defeated, but he, like Link, was also blessed by the goddesses — the Triforce of Power protected him — and could not be executed. Instead the Sages who guard the Mirror of Twilight used it to banish him to the Twilight Realm.
Precipitating Twilight Princess
Amongst the Twili was one named Zant, a servant of the Royal Family (of which Midna was a member). He was greedy and over-ambitious, and became embittered by the Royal Family's lack of action with regard to what he saw as the Twili's pitiful situation. At this point he met Ganondorf, and made a "deal with the devil" with him to receive great power in exchange for his servitude.
He then overthrew Midna and turned her into an imp, and the rest of the Twili into Shadow Beasts under his dominion. Zant, acting under Ganondorf's orders, cast the world of light into shadow and proclaimed himself the "King of Darkness". However, when he later tried to shatter the Mirror of Twilight he could not completely destroy it — Midna calls this the proof of his false kingship.
Twilight Realm
Despite the hyperbole that Zant uses to talk about the Twilight Realm and the suffering which has been inflicted on the Twili — "[We] were locked away like insects in a cage. In the shadows we regressed, so much so that we soon knew neither anger nor hatred... nor even the faintest bloom of desire." — Midna herself seems fond of the Twilight Realm. She says, in an early part of the game, that she feels more comfortable in the twilight-stricken areas of Hyrule — "The black clouds of twilight are so fetching today... I feel so much more at ease here."
The actual language that Zant uses is important. He says that the shadows of the Twilight Realm stripped the Twili of both anger and hatred, and of desire. Assuming that by 'desire' he refers to the burning to possess the Sacred Realm that led them to be banished in the first place, this almost seems like a blessed existance.
Midna says that the reason he was not chosen as the lawful leader was that the Twili recognised in him the greed that had them consigned to the shadows in the first place. Zant himself says that it was "in the thrall of hatred and despair" — these very emotions which the rest of the Twili supposedly no longer felt — that he encountered Ganondorf. And in fact, it was Zant himself who regressed the Twili into imps and Shadow Beasts, using the magic that Ganondorf had granted him.
Although Midna refers to the denizens of the Twilight Realm as "doomed to live in the twilight, flitting in the half-light of dusk," she also says that her world was a peaceful place before Zant rose up to seize control. Rather than a punishment or a prison, in some ways the Twilight Realm seems to be a rehabilitation from the goddesses: a place that, though they can't leave and return to the world of light, the Twili could find peace from the anger, hatred and greed that drove their attempt to capture the Sacred Realm.