Monsters are creatures that modern day science doesn't believe in. They are mostly described as villains of epic stories in which demigod-heroes will happen slay a mighty beast. However, mortals simply just refuse to believe that they are real, and therefore they do not see them. Some mortals, such as Rachael Elizabeth Dare (in the Percy Jackson series) can see through this barrier, or this mist, the magical veil that hides human vision.
Some monsters are ancient creatures that would terrorize, fight, and kill half-bloods (and humans too, before the mist was created).They are soulless, killing archetypes that whenever defeated, they will end up in Tartarus. (Tartarus is the deepest, darkest, worst part of the Underworld.) So, monsters can never truly die, which really sucks, but they can be disabled for a lifetime (if the hero is lucky) or they can fade.
There are several species of monsters so intelligent and so powerful they are characterized under being a Race. This means they have they're own culture, history, government(this term used very loosely). Unlike most monsters they usually do not go to Tartarus but have they're own place that they go, or they are reborn into something.
Many Races can be found working together or going to war agaisnt eachother. Almost all have a patron either a human god or they're own unique deitiy.