A coat of arms is a design used to identify a particular family.
The notion started in the Middle Ages, when knights in battle could only be distinguished by friends and foes by the design on their shields - everyone looks the same in a suit of armour, especially if they are waving a large sword at you!
The coat of arms would be made up of designs that reflected the family status, and would be passed down through the generations, which is why only laded or titled families have them these days.
More here -
http://www.historicalnames.com/origins.htm