Public quests are quests that everyone in the area can participate in at the same time. For instance, in the first public quest for Chaos, "Ruinous Powers", the Sorcerers of Tzeentch are performing a dark ritual to summon a great beast of change. To finish the quest, players need to kill 25 meddling Thorshafn Militiamen. Each Militiaman that is killed advances the counter. Your reward when the quest is completed is based on how many of the 25 Thorshafn Militiamen were killed by you.
Many of the public quests are done in stages. For instance, in "Ruinous Powers", once you have killed the 25 Thorshafn Militiamen, players will need to harvest souls from tombstones to bolster the power of the dark ritual. When you've harvested the correct amount of souls, a demon of dark power emerges, and you must prove the Raven Host's supremacy over all other ruinous powers by defeating the denizen of rage.
When you enter an area where a public quest is underway, a message is displayed in the center of your screen. Progress of public quests is displayed in the top right of the UI.
Things you do in WAR have an influence on the world. How much of an impact you make is entirely up to you, and you can influence each area you travel to in a different way. This is mainly done by participating in Public Quests, and if your influence rises high enough in an area, you can earn special gear.