The concept of gemstones having special powers was first noted in the Bible, in the Book of Exodus, where the Breastplate of Aaron is described as containing 12 special stones.  Auccording to scholars, in the 1st and 5th centuries AD, these 12 stones were associated with the zodiac.  People would wear stones according to the qualities they wanted to possess.  In the ancient art of Vedic Astrology, stones were linked to the 9 planets, and were chosen to enhance particular positive qualities or to offset negative aspects of someone's natal chart. 
Our modern day use of birthstones, where each month is represented by a stone, dates back to the 18th century Polland.  The particular month stones were not set in stone (pun-intended) until 1912 by the National Association of Jewelers.  They have remained unchanged except for the addition of Tanzanite to the month of December in 2002.  
In general, birth months have more than one stone.  One considered precious, and one or two semi-precious stones.  Birthstones go by the month one is born, as opposed to their sign of the Zodiac.