The use of coats of arms dates back to the Middle Ages. Originally, coats of arms were granted to knights and were displayed on their armour. Eventually, the number of coats of arms and the potential confusion among them led to the development of an official class of heralds, who regulated such matters. The study of coats of arms is called heraldry.
A "coat of arms" is sometimes also correctly referred to as a "heraldic achievement" or "armorial bearings". Many people also use the terms "coat of arms" and "crest" interchangeably. However, a coat of arms is not the same thing as a "crest". A crest is just one part of a coat of arms. Specifically, it is the part of a coat of arms that appears at the top, above the shield. There are books of family crests, but these books will not show you a family's full coat of arms, just the "crest" part of it.
There are two ways to legitimately acquire a coat of arms. One is to establish a hereditary claim, and the other is by a new grant.
If an ancestor of yours was granted a coat of arms, then you may have a hereditary claim. However, it is important to understand that there is no such thing as a "family" coat of arms. A coat of arms is granted to an individual, not to a family. It can be passed to a person’s descendants, through the direct male line. However, the descendants add what are called "difference marks". Difference marks are standardized. For example, an eldest son adds a difference mark called a "label" to his father’s coat of arms. A second son adds a crescent, a third son a mullet, and so on. There are a total of nine standard difference marks. After the father dies, the eldest son can use his coat of arms without the label.
If you cannot establish a hereditary claim to a coat of arms, you may be able to have a coat of arms created for you. Of course, you can create your own, or have one for you created by a commercial outfit, but it will have no official status. To be an official, legitimate coat of arms, it must be created by one of the heraldic authorities. These authorities have been delegated the right to grant arms originally exercised by the monarch in medieval times.
Images - 1. Arms of Her Majesty in Right of Canada, 2. The Crest from the Arms of Her Majesty in Right of Canada.