The traditional way of counting age is like this:
- You are 1 years old (한 살) when you're born.
- You gain a year on January 1st.
So technically, a baby born in December will become 두 살 - two years old - on January 1st. However, for children that young, they normally count their age by days, weeks or months (just like in English).
Also, sometimes age of babies can be expressed using the word "돌". This normally means the first birthday of the child (e.g. 돌잔치 - first birthday celebration), but can also be used to say the baby is (about) 1 year old (in the Western sense). So for a 12-month-old baby you can say:
그는 벌써 한 돌 됐어요.
It can even be used for an older child, though normally just babies:
우리 아이는 이제 겨우 두 돌이 넘었다 (Our baby is now barely 2 years old)
(The last example is taken from the 표준국어대사전).
Also, more and more Western age is used. This is expressed with the phrase 만 X세. So 만 6세 means 6 in the Western sense. This is often used with regulations related to children. There's even a calculator that helps you calculate "만 나이".