Although what Alan said is correct, I want to comment on the 9 month/10 month thing. Neither 9 nor 10 months is accurate. 10 months is accurate if you view 4 weeks = 1 month, but obviously that is not usually the case. It's actually somewhere between 9 and 10 months. This is not just something Korean people think (I've heard many mothers in America say 10 months as well, even though 9 months is what people normally say). It's because a full term pregnancy is 40 weeks, and will usually span across 10 months if it's a full term baby. Women who are actually pregnant will go by weeks rather than months anyway. Since there are about 52 weeks in a year, the ~40 week pregnancy is rounded up to conclude that babies are 1 year at birth, since 40 weeks is much closer to 1 year than to 0 years.
Also some additional info, I asked a Korean friend once and she said that in Korea they too count baby's ages by months (5 month old baby, 13 month old baby, etc). It's just not their official age. So while a newborn with a bday of December 31 would be considered 2 years old, they would also be considered 1 month and etc. This is probably just for moms to be able to tell their baby's progress, and just like with us, they stop using the month counting system after 24 months or so.
This isn't really pertinent to your question, Alan already covered all of that, I just felt like adding on to some of it.