A cornucopia, also called a horn of plenty, is by tradition made of a goat's horn filled to overflowing with fruits, flowers and grains of the harvest. It symbolizes prosperity, abundance and good fortune. It also has associations with fertility.
Origin of the name
The name comes from the Greek, “cornu” meaning horn and “copia” meaning abundance or plenty--thus literally, “horn of plenty”.
In Greek mythology
One version of the origin of the cornucopia begins with Zeus, who as a young child was suckled by a she-goat named Amalthea. When he accidentally broke off one of her horns, he gratefully returned it, first endowing it with the ability to be filled with whatever the holder desired. Zeus later placed Amalthea in the sky where she became the constellation of Capricorn, the Goat.
Another version has Hercules fighting a bull (actually the disguise of his rival, the river god Achelous) in one of his twelve labors; after tearing off a horn he gave it to the naiads, who filled it with flowers. It was later acquired by Copia, the goddess of plenty.
As a Motif in Art
The cornucopia is frequently depicted in classical art. The never-empty horn of plenty was associated not only with Zeus and Hercules, but with other mythological deities, especially Tyche, the goddess of good fortune and prosperity. If the horn was facing upward, it symbolized Dionysus, the god of wine and festivity, and Demeter, the goddess of the earth, agriculture and fertility. 
As a Religious Symbol
David Hendon, in an article in the The Journal of Ancient Numismatics, says that the cornucopia was a very popular religious symbol in the ancient world. It can be found stamped on Jewish coins beginning in the Maccabean era, and can also be found on objects like seals and rings, and in architecture. This was during a period of Hellenistic influence, when many civilizations were influenced by the culture of the Greeks. In the Jewish world, where animal horns were used to anoint kings and for religious musical instruments, the cornucopia took on added significance. 
As a Symbol of the Harvest
In traditional European cultures, communities celebrated the completion of the harvest by creating a corn dolly or corn maiden made of straw. These "dollies", or idols, were intended to ensure fertility and prosperity in the year to come, and took many shapes. One of the earliest forms used was that of the horn or cornucopia. 
Precisely when the horn of plenty made its way into our national consciousness as a Thanksgiving holiday emblem is hard to say. But with its message of abundance and good fortune, the cornucopia is now a symbol of Americans’ sentiments of thankfulness each November.
Cornucopias have become traditional autumn centerpieces and usually feature not a horn, but a basket woven into a conical shape. Modern day fillings might be leaves, dried or fresh flowers, cattails, nuts, and of course fruits and vegetables such as grapes or miniature gourds and pumpkins.
Whether it goes by the name cornucopia or horn of plenty, the meaning of this ancient symbol still resonates today. The reverence people feel for the fruit and plenty of the earth remains unchanged throughout the centuries.
Harvest Home, The Autumnal Equinox by Lois Tilton
 David Hendon; The Cornucopia Served As a Jewish Symbol
 M. Lambeth; Discovering Corn Dollies; 2008.