This article is from the rec.heraldry FAQ, by Gordon Findlay (firstname.lastname@example.org) and Francois Velde (email@example.com) with numerous contributions by others.
This is a difficult question to answer; it requires a great deal of
research and skill.
In most countries in the world, you can bear any arms you want. This is the
way in which arms were originally adopted, before codification and
regulation by European heralds and rulers.
However, many people consider it wrong to adopt someone else's arms. In some
countries, notably Scotland, this is not only dishonourable but illegal.
In particular, there are no laws regulating the use of _non-governmental_
arms in the US. The American government neither grants nor recognizes
armory. You can adopt any arms you choose and use them however you want
(unless you infringe on someone's trademark, which is an entirely different
subject that has been beaten to death on this newsgroup so please don't ask
about it); but you have no particular right to those arms or any other.
If you are descended from someone who was granted arms by some heraldic
authority then you may have some claim to those arms within the
jurisdiction of that authority. The chances are very good that you do not
have any claim on any actual arms. Most people in the world do not.
Exactly what conditions you have to meet to establish such a claim vary
considerably from one country to another. At the very least, you will have
to prove that a recognized holder of the arms is your ancestor. In some
countries, you would have to prove that you are the legal heir of that
person. Getting an official recognition of your claim is likely to be
expensive and time-consuming; in England, for example, it costs thousands
*** Your last name has nothing to do with the matter. ***
Arms are not associated with surnames, but with individuals and, in some
countries, with families. The important thing is who your ancestors are,
not what surname you happen to bear. The fact that your name happens to be
"Smith", for example, gives you no claim whatsoever on any of the thousands
of arms borne throughout history by various people named "Smith."
Unfortunately, there are lots of unscrupulous businessmen worldwide who are
happy to promulgate false information about the subject of armory. They will
happily take your money to tell you "Your Family Arms", which they supply
simply by finding an armigerous family that happens to share your surname.
We suggest that you avoid these companies; if you want anything more than a
decorative wall-hanging, they are a waste of your money. And if you will
be happy with any pretty picture to hang on your wall, you can save
yourself the trouble of dealing with these companies, and simply choose
arms that you like.
See also the MFAQ at: http://www.heraldica.org/faqs/mfaq