When the child is first born, Irene will put a baby bed in your house. Your child will live in its bed for 2 seasons. You can pick up the baby and even play with it by using the Baby Rattle, which you can buy from Barbara's store.
The child will grow to a new stage every 2 weeks until he/she is grown up. There is the baby stage, the crawling stage, the walking stage, and then the talking stage. Each time your child changes stages you'll be asked the personality question.
In the Japanese version, at the talking stage you have the choice of name you want the child to refer to you as. There's casual name like Daddy and Mommy, and then more respectable ones like Father or Mother. This doesn't appear to occur in the English version.