Demigods are half mortal half god. They can be born from the gods Roman or Greek counterpart. They are born when a human and a god share sexual relations. Another term for demigod would be Half-Blood.
Demigods are the result of a human and a god making a child. Because they are part god, they have better reflexes and are hardwired for a certain language. Greek demigods for example understand Ancient Greek, while Romans understand Latin. Mortals see their abilites as Dyselxia (they have a hard time reading languages outside of Latin and Ancient Greek), and sometimes ADHD, because they see too much, which can save their life in a battle. They have a dominant control or skill over the realm of their godly parent, as they inherit some of their power.
When a demigod reaches a certain age, normally 10, 11, or 12 they begin to start realizing who they are. This enhances their scent for monsters to catch, and attack them. They have to get to either Camp Half-Blood (if they are Greek demigods) or Camp Jupiter (for Roman demigods) to finally be safe, and be among their kind. It is rare that a demigod does not go to camp, but this has been demonstrated by Annabeth Chase and her trio (Luke Castellan and Thalia Grace) when they lived on their own escaping from monsters with various safe houses.
Few demigods actually have full blooded siblings, unless they were raised together, or are twins. Normally, you are a child of the same god, but not the same mortal parent. Demigods often go on quests to fight any evil uprisings, or as a little challenge. Rome takes it more serious than the Greek's only issuing quests after they have a prophecy, while the Greek's send their demigods out for small accomplishments (and bigger things). Greek demigods are issued quests by the Oracle of Delphi, and Romans get a prophecy from a Roman god or Octavian the augur.