The Trust system is a fairly unique feature in Binary Domain.
More or less a Consequence System, "Trust" is something that you will earn (or lose) during the game. Each squad member on your team has a different personality, so he or she will response differently to your actions. Something one squad-member may agree with (charging the enemy line), may cause another member to question your command decisions. Combat prowess, voice-commands issued and responses during conversations between Dan and a squad member all affect Trust levels.
Trust levels can affect the story of Binary Domain and your squad's overall effectiveness. It can also slightly alter the ending. Reaching the maximum Trust level with any squad member will unlocked that respective squad member's achievement.