Influence points are used by clicking on another Sim and choosing the influence menu. A list of things to influence a Sim to do will highlight which ones are available. Examples of what Sims can be influenced to do include: clean the house, repair broken objects, do homework, and my favorite, write term papers.
I have found little use for the influence system in the regular neighborhood. In college, I use it to make NPCs students write term papers for my Sims. You are not required to use the system and ignoring does not hurt.