If you ever go to law school, you will become familiar with this mantra:  Duty, Breach, Proximate Cause, Damages.  First year law students learn that, before a person can be found guilt of negligence and held liable for another person's injury and damages, it must be shown that:
 
(1) the defendant owed a duty of care to the plaintiff; (2) failed to meet (breached) that duty of care; (3) his breach of duty caused plaintiff’s injury; and (4) the plaintiff suffered damages as a result of that breach of duty.
 
 Your question about the "reasonable person" comes up in the analysis of number 2, above.  The terminology may vary from state to state, but the finder of fact (judge or jury) will answer the question of whether the defendant breached his duty to the plaintiff by asking another question:  Did the defendant act as a reasonably prudent person would under similar circumstances?  
 
For example, let's say A sued B for damages he suffered when he was walking down the street and was struck by a mattress that B was carrying on his roof rack without any ties or rope to hold the mattress in place.  The jury would ask whether a reasonable person would have secured the mattress (and would likely answer that question in the affirmative.)  Hope that helps.