How do Chinese people traditionally count their age? I remember from a children's book I read long ago they did it differently and they would end up being a year younger then they say. Why do they do this?
When you say 'Chinese people', bear in mind that it includes about 1.4 billion sods, some of whom have very different cultural outlooks indeed. The vast majority of Chinese people these days count age the same way as anywhere else in the world, just have a look at their identity cards (yes, all four major Chinese jurisdictions have those - that is Mainland China, Taiwan, Hong Kong and Macau) and you'll find it has a date of birth, so their age is counted like anywhere else.
However, your book is not wrong, either. Traditionally, Han Chinese counts the 10 month pregnancy towards a person's age, so a newborn is counted as one year old. which is roughly translated as 'false age'. Under this system, one's age increments by one year every Lunar New Year, instead of during one's birthday. It is possible for a child of a month old to be reckoned two years old this way, if it was born a month before Lunar New Year. This is known as 'xuling 虚龄'. That way it is possible for a person to report his age as being up to two years older than his actual chronological age.
The normal method of counting exact chronological age is also used in traditional society, it is known as 'zhoushui 周歲', or cyclical age.
In some places, the concept of 'Nanjin nuuman 男进女满' is applied, where zhoushui is applied to females and xuling is applied to boys.
However, this Chinese person has not heard of people using xuling outside of literary or jocular context from people born after 1940, and even those who do typically know when they are actually born and how old they are chronologically.