Coat of arms
Heraldry is said to have its origins in battle identification of individual knights. This identification was usually represented on the shield. Later it continued as a visual representation of the person, and continued to represent just that one person.
First, the legal point of view. Like other property it was inherited by the heir, and that continuity of use is often misrepresented into thinking of it as a family 'possession'. There are courts which control that possession. In Scotland it is the Court of the Lord Lyon.
There is no such thing as a 'family coat of arms', unless you think of it as a piece of property passed from one person to another. If anyone else in the family wanted to bear arms, they would distinguish them in some way with another mark or symbol.
So here is a representation of a coat of arms used by a succession of Ralstons of Ralston, representing the head line of inheritance.
Blazon: Argent on a bend azure three acorns in seed or proper; meaning on a silver shield a blue band stretching from the top left corner to the bottom right, on which are drawn three gold acorns. The blazon is the heraldic description / language which is a mixture of Old French and English words.
Motto: Fide et marte; meaning By Faith and By War
Some members of the family have used a coat of arms in this incorrect way. Alexander McDougall Ralston of Warwickhill died in 1862. His headstone in Dreghorn Churchyard includes the Ralston of Ralston coat of arms. He never matriculated (registered a variation) with something which made his shield distinctly his own.
The term 'family crest' is also widely misused. It is not the same as 'coat of arms'. Strictly speaking the crest is the object on top of the helmet. In this case it is a falcon.
The clan system in Scotland often uses the crest in the form of a badge for use by members to signify membership.