In each chapter there is at least one Public Quest. By entering the area of this Public Quest you are given Public Quest Objectives, these can be to kill mobs or capture objectives. By participating in the Public Quest and aiding with the tasks of the objectives you are granted Influence. You do not have to be in a group to share Influence points with other players focusing on the same objective as you are, the amount of influence you are given is calculated by how much you aid on completing that objective. Additionally, you will also gain Influence by helping the Public Quest move on to the next of its three parts. If the Public Quest is completed you are given an Influence bonus depending on how much you have aided to the success of the Public Quest (Damage, Healing, Time Spent).
The Influence Points you earn are stored on the Influence Bar on the top right side of the screen, next to the Mini map. The Influence Bar is very similar to the Experience Bar or the Renown Bar with one exception. On the Influence Bar are three silver marks (Gold once reached). These three marks indicate when you have earned enough Influence to collect your rewards. If you want detailed information on how much Influence you have, how much is left for the next mark and how much is left for the last mark, you can click the Influence Bar which will bring up your Tome of Knowledge. Once in the Tome of Knowledge you will see an Influence meter much similar to the Influence Bar. By hovering over this meter, you will get the detailed information on your Influence progress in numbers. Here in the Tome of Knowledge you will also get details on what rewards you will get and the lore behind the Public Quests if you have unlocked it.
When you complete Kill Collector quests (having killed so many mobs that you won't get extra xp for more of them), the game will erronously tell you that you received an influence reward. This is a leftover message from an old system, and no longer applies.
To collect your rewards visit the Rally Master in that chapters specific town. If you do have a reward to collect, the Rally Master will have an turquoise coloured icon indicating that. There is a predetermined template on what kind of rewards you will get from reaching each of the three parts of the Influence Bar. By reaching the first mark, Basic, you are given a minor reward, this is mostly a choice between one of several potions with its quality set to match that of the Zone. The second mark, Advanced, gives a choice between two lesser armors, weapons or accessories usable by your career and race. The quality of these varies from common to uncommon by Tier. Lastly the final mark, Elite, gives you a choice between two greater armors, weapons or accessories also usable by your career and race. The quality of the Elite rewards varies from uncommon to epic depending on tier.