Conquest battles are an important part of establishing dominance in the Peloponnesian War. The battles are available as part of the main storyline and as side missions you can pick up when exploring a region.
Whilst you may not be able to pick which nation you support during the story quests, you can lend your mercenary skills to either side when you choose to take part in conquering a region during your travels.
To take part in a Conquest battle, you must first lower the power of a nation currently in charge of the region.
This will uncover a Battleground on the map and with it, two Battleground camps you can visit to offer your services to either Sparta or Athens.
Speaking to the commander of the nation you wish to back up will initiate the Conquest battle. The difficulty of each battle, as well as its rewards will vary depending on which side you support.
To win the conquest battle, seek out Captains and Heroes during the fight to significantly lower the enemy army strength. Although killing regular soldiers also works, you may find that the fight will take longer.
Reducing the enemy army strength to zero results in a victory for your nation, allowing them to seize the region and assign a new Leader.