A painted shield is one rendering. The coat itself, for the record, is the blazon: a short description from which a competent artist can draw the arms again. Encyclopaedia Britannica traces the word to French blason. It once meant the shield. Today it usually means the verbal description, and there is a verb as well: to blazon.
Every formal blazon begins with the field — the background of the shield. Britannica’s account of heraldic design puts that layer first, then the charges that lie on it. The field may be a metal, a colour, or a fur.
The metals are or (gold, often drawn as yellow) and argent (silver, often drawn as white). The ordinary colours are gules (red), azure (blue), vert (green), purpure (purple), and sable (black). The common furs are ermine and vair. Britannica notes two rarer British stains as well, murrey and tenné, which you will meet less often in a simple family shield.
A compact example from that same grammar: sable a chevron ermine between three lions rampant argent crowned or. The field is black. The chevron is fur. The lions are silver, they stand on their hind legs above and below the chevron, and their crowns are gold.
Heralds keep contrast so a shield can be read at a distance. The old rule of tincture tells the painter not to put metal on metal, or colour on colour. Gold lions on a white field break the rule. Gold lions on a red field keep it. Furs are a middle case and are often allowed to touch either metals or colours. The rule is a habit of good work, not a law of physics, and later arms sometimes break it on purpose. For reading a new design, it is still the first test.
The Lord Lyon’s notes on blazoning remind the reader of another Scottish fact while they teach the words: there are no “family coats of arms” in the souvenir sense. Arms are individual. People of one surname may share a family resemblance in their shields, built on similar undifferenced arms, but each recorded coat still belongs to one person.
Dexter and sinister are the bearer’s right and left, not the viewer’s. When you look at a shield, dexter is on your left. That is why a lion facing dexter appears to look toward the left edge of the page. Positions of beasts — rampant, passant, couchant — are part of the blazon, not decoration the artist may change at will. The Notre Dame heraldic dictionary is a clear public walk through those parts of a full achievement once you have the shield itself.
If the question is whether a surname “has” a coat at all, begin with The crest is not the coat of arms. If the question is what an American may lawfully assume, see Heraldry in the United States.