The Mist is a type of magic or force that stops mortals from seeing monsters, gods, titans and various other mythical creatures and supernatural occurances.
The Mist replaces these things with visions of (relatively) ordinary happenings. Under the influence of Mist a mortal would see gods or monsters as ordinary humans or animals, and supernatural objects also become commonplace items. Some animals, such as rats, and some very rare mortals and small children (but their parents never listen to what they say) can see through the Mist. Half-Bloods and monsters can all see through the Mist at most times (though sometimes the mist is strong enough to fool demigods) and manipulate it to a certain degree, only by certian people. Gods are completely immune to the effects of the Mist.
According toChiron, the Mist is mentioned in the Iliad. It is not specified which part he is referring to, but Book XX of the Illiad mentions Poseidon pouring our mist over the eyes of Achilles to protect Aeneas. Book V also specifically describes Athena giving Diomedes the ability to see gods:
"And I've removed the filter from your eyes, which covered them before, so now you'll easily distinguish between gods and men"
The mist can be altered by certain powerful demigods. Two of these demigods, Missy (Daughter of Hades) and Perceus (Daniel; Son of Poseidon) became the God and Goddess of the Mist. They now regulate it, and personally deiced who can see through it and who cannot.