History of Coats of Arm and Rules of Heraldry
The Coat of Arms became best known as a means of identification of knights during tournaments.  These knights wore suits of armor and displayed their Coat of Arms on their shield, banners, and elsewhere.  When knights arrived at the tournaments, he would blow a horn to announce his arrival and in response heralds would describe the knights coat of arms out loud.  The term blazon indicates the act of describing the coat of arms out loud in response to the blowing of the horn.  Today, blazon simply means describing the coat of arms in heraldic terms.  As the popularity of tournaments grew, the Coat of Arms became highly desired and was granted to wealthy and prominent families.  Heralds continued using their expertise in order to identify families who displayed their Coat of Arms presumably so those families may be taxed.  

Strictly following the rules of heraldry, family members were only allowed to display Coat of Arms where they could make an official connection with the original recipient of the COA as agreed upon by The College of Arms heralds.  During these ancient times, the Coat of Arms was so highly regarded that Visitations occurred in which the College of Arms heralds would go to households where individuals had to demonstrate a connection with the original COA bearer in order to have permission to display the same COA as their own.  This could be done by showing the original grant of arms or by genealogical records showing the proper lineage.  Those individuals who were unable to prove their right to arms were expected to publicly denounce their coat of arms as a legitimate granting of arms.  Some people were sued in English courts for displaying false Coat of Arms.

The Right to Arms can be granted to males (or females) in the line of descent who can prove a connection with one (or more) ancestor (s) who received the Coat of Arms.  Some families used COA without having the actual right to arms (a confirmation did not occur).  Similarly, some families used COA where a COA for that family did not even exist.  It is not uncommon for families to use a COA where it does not appear in the College of Arms records (even today).  The Coat of Arms was the family trademark and was placed on flags, jewelry, clothing, and elsewhere.  For more information on heraldry go to the following Wikipedia page:  http://en.wikipedia.org/wiki/Heraldry

Proving The Right to Arms
The process of proving the Right to Arms is outlined on The College of Arms website.  In summary, individuals wishing to prove their right to arms should contact the College of Arms either by letter or by their website. Once contacted, a herald will be assigned to a persons case based on weekly rotations.  The herald may run a name search (upon payment of fees) in order to locate armigerous families of the same name.  Once a coat of arms is located, the client should produce an ancestors pedigree such that it can be connected with an existing pedigree on file at the College.  After a connection is made by the herald, the client must produce full genealogical records connecting himself/herself to the ancestor on file.  The genealogical records are reviewed and once approved, the client may be granted the Right to Arms.  See more details on The College of Arms at their website:  http://www.college-of-arms.gov.uk/About/09.htm