To find out if the person is breathing, lie the person down flat on their back and listen through their mouth/nose to hear if they are breathing or look at their chest to see if it is rising. Here are some reasons why they may not be breathing and what you can do about it: 

1) they choked on something - perform the Heimlich maneuver (http://www.youtube.com/watch?v=kJDpr05zmB4). 

2) their brain stopped telling them to take breaths - if you can, push air in side their lungs for them, use a barrier device when possible (e.g. pocket mask)

3) their lungs are filled with something else - tilt the person on their side to "pour" some of the fluid out for a few seconds 

To find out if the person's heart is beating, you can learn to check their pulse. You can read through http://www.wikihow.com/Check-Your-Pulse orhttp://theemtspot.com/2010/05/22/the-ultimate-emt-guide-to-vital-signs/. Pick a couple places (usually carotid and radial) and learn how to check pulses in those places well. You do not have to know the rate. 

You should always assume the person does not have a pulse unless you can prove otherwise. Therefore, if you don't know how to check for pulses or if the person does not have a pulse, immediately start pumping their heart for them. Here is a video on how to do that while also breathing for the person until professional help arrives: http://www.youtube.com/watch?v=IUB0Ep4DuBI 

Stay tuned on what you can do if someone is unconscious after an accident or trauma!