The idea that certain precious stones were associated with certain months goes back at least to the 1st century CE.  Oddly enough, the idea of wearing one's birth stone - that is, wearing a piece of jewelry containing the gemstone associated with the month one was born in - appears to be much more recent.  George Frederick Kunz, in The Curious Lore of Precious Stones, states that the custom began in Poland during the 18th century.

The wearing of one's birthstone is traditionally thought to be lucky.  Thus, if one was born in the month of January, and one wears a garnet (one's "birth stone"), one should attract good fortune.