Demigods, or half-bloods, are a race of metahumans who are half-mortal, half-god. They possess mortal souls and are vulnerable to old age and death; however, their godly blood endows them with special abilities that allow them to achieve feats usually not possible by humans.
Greek demigods, such as Percy Jackson, are a subset of demigods who train at Camp Half-Blood, located in Long Island. They are born when a god conceives a child with a mortal while in their Greek aspect. As such, Greek demigods have an innate predisposition towards ancient Greek, and possess instinctive battle skills that manifests as ADHD.
They are assigned to a cabin dedicated to their divine parent once they are claimed. Most Greek demigods die in their teens because they are prone to being attacked and killed by monsters before making it safely to Camp Half-Blood. Very few survive to make it past their twenties.
Roman demigods, such as Jason Grace, are a subset of demigods who train at Camp Jupiter, which is located near San Francisco. They are born when a god conceives a child with a mortal while in their Roman aspect. They naturally speak and read Latin, as opposed to Ancient Greek to Greek demigods, and are trained in the style of the Roman legions instead of Greek warfare. Roman demigods can live in New Rome and start families.
Roman demigods start their lives at the Wolf House. It is there that Lupa decides if the child is strong or not. If she believes they are, she will spend time training the demigod skills that they will need to survive. Should the demigod make it through the training alive, Lupa will instruct them to follow their instincts to Camp Jupiter, home of the Roman Legion. However, if Lupa doesn't believe the demigod is strong enough or shows fear, it is said that they will be eaten by her and her wolves. Upon making it to Camp Jupiter, a praetor, currently Reyna will decide if the demigod can join the Legion before sending them to have their augury read. If it is favorable, the demigod can then officially join the Legion. Like the Greek demigods, they were unaware of the existence of their counterparts at Camp Half-Blood in Long Island, but some demigods, like Octavian had their suspicions.
This is due to a conflict between the two groups back during the American Civil War (and likely many other wars before), after which the gods manipulated the Mist as tightly as possible to stop the two groups from ever meeting each other again, although there have been some close calls. The head of their camp are the camp's two Praetors, who are elected demigods from the camp.