A birthstone is a gift of a precious material (jewelry, mainly gemstones; themselves traditionally associated with various qualities)
that symbolizes the month of birth (in the Gregorian Calendar)

It is sometimes also called birthday stone (cf. infra; but that word is, confusingly, sometimes used as a synonym for an anniversary gift, which is related to the recipient's age, that is, year of birth).

There have been many different sets of birthstones used throughout history and in different cultures. In 1912, in an effort to standardize them, the American national association of jewelers, Jewelers of America, officially adopted the following list; it is currently the most widely used list in the United States and many other locations, including Australia and Thailand. Some alternates have been adopted to be a less expensive substitute for a cut stone.

Month	Modern Birthstone[2]	Alternate	Traditional Birthstone[3]
January	Garnet	Rose Quartz	Garnet
February	Amethyst	Onyx	Amethyst
March	Aquamarine	Red Jasper, Bloodstone	Bloodstone, Jasper
April	Diamond	Rock Crystal (Quartz)	Diamond, Sapphire
May	Emerald	Chrysoprase	Emerald, Agate
June	Pearl, Moonstone	Alexandrite	Alexandrite, Emerald
July	Ruby	Jade or Carnelian	Ruby, Onyx
August	Peridot	Aventurine, Sardonyx, or Sapphire	Sardonyx, Carnelian
September	Sapphire	Lapis lazuli	Sapphire, Peridot
October	Opal	Pink Tourmaline	Opal
November	Yellow Topaz	Citrine	Citrine, Topaz
December	Turquoise	Lapis Lazuli or Tanzanite	n/a

Tanzanite was added to December by the American Gem Trade Association in 2002. Most organizations do not recognize
tanzanite as a December birthstone, however, and the AGTA's move to make it a December birthstone
has generally been viewed as a marketing ploy.

The birthstone seems to originate from Biblical times. The Breastplate of Aaron, referred to in Exodus 39:10-14:

10 Then they mounted four rows of precious stones on it. In the first row there was a ruby, a topaz and a beryl;

11 in the second row a turquoise, a sapphire and an emerald;

12 in the third row a jacinth, an agate and an amethyst;

13 in the fourth row a chrysolite, an onyx and a jasper. They were mounted in gold filigree settings.

14 There were twelve stones, one for each of the names of the sons of Israel,
each engraved like a seal with the name of one of the twelve tribes.

The precise list of birthstones however can be found in Revelation 21:19-20 where the foundation stones of the
new Jerusalem are listed, in the order of the Roman calendar:

19 And the foundations of the wall of the city were garnished with all manner of precious stones.
The first foundation was jasper; the second, sapphire; the third, a chalcedony; the fourth, an emerald;

20 The fifth, sardonyx; the sixth, sardius; the seventh, chrysolyte; the eighth, beryl; the ninth, a topaz;
 the tenth, a chrysoprasus; the eleventh, a jacinth; the twelfth, an amethyst. ==

Information courtesy of  wikipedia.org