The more experience points a character gains when killing a monster in Vanadiel, the more Conquest Points they will receive. The base Conquest Point amount gained is one-tenth the amount of experience gained, with bonus amounts given for defeating monsters in regions controlled by other nations or the beastmen. This means that a solo character who kills an Even Match monster will gain 10-20 CP. You can see how many Conquest Points your character has by pulling up the Regional Map (either via the main menu, or with Ever Quest 2 Gold command); the characters CP will be listed in the bottom right corner.
A character will only recieve Conquest Points if they have a signet on when killing monsters. Signets are given out by three different types of guards:
National Conquest City Guards at the gates of the home city of the character.
National Conquest Foreign Guards located at the border and outposts in regions that a characters home nation controls, as well as embassies in foreign nations.
National Conquest Jeunoan Guards found in Jeuno.