That seems like a simple enough question. But the answer isn't that easy. It depends on who you ask (and maybe what they are selling!). Different cultures at different times have adopted different versions of what stones go with which month. In ancient India the December birthstone was Ruby. Among the Mystics of Tibet it was Onyx. In western societies the traditional December Stone was Turquoise or sometimes Lapis Lazuli. In more recent times it was declared by the Jewelers of America to be Blue Zircon. In 2002, the American Gem Trade Association added Tanzanite as a Birthstone for the last month of the year. Some listings also include Blue Topaz as an option. Discounting the ancient Indians and Tibetans, the common denominator is that it is something blue. If none of these work for you, there are also birthstones assigned for the zodiac symbols and for the days of the week!
The bottom line, to me, is that it doesn't matter. Some people feel obligated to wear stones or give somebody stones based on their birth month. Buy or give gemstones because you like the way they look or make you feel or because they go with what you are wearing. Don't worry about what AGTA or your jeweler or some Tibetan Monk says you should wear.
Chart of the Traditional & Modern Birthstones