Arms are property. The first person to carry a coat of arms passes that coat to his descendants, who pass in on to theirs, and so on. Different countries have used different rules for the transmission of heraldry. Traditionally the right to bear arms has been transmitted through the male line only. In some places, all sons of an armiger (someone who bears arms) inherit the arms; in others only the oldest male inherits the arms and all other sons must change the arms in some way to show that they are a junior member of the family. (British law was recently changed to allow women to inherit arms, but this is such a recent change that it will not affect genealogical research).
Thus, finding arms attributed to your surname does not mean that you have the right to carry those arms; it means that someone with your last name once had the right to carry them. The same surname is often borne by unrelated people, and you have to show a direct male line of descent to a person who is known to have arms. Demonstrating this relationship can be a major task.
For SCA purposes, it would be appropriate to find arms belonging to a person with the same surname and period as your persona and to difference them in some way. Almost any change in arms was used as differencing; among the most common are adding a set of charges, changing a tincture in the arms, or changing the type of a secondary charge.
If you're looking for sources about genealogy, a good place to start is the Yahoo index on genealogy.
For real-world heraldry, check Francois Velde's Heraldry Site. For specific information about English heraldry, you can contact The English College of Arms.
Last modified 30 August 2001.