That's a Contact form.

You can find out how to modify that form from just about any Outlook Book.
The basic procedure is here:
http://office.microsoft.com/en-au/outlook/HA012106101033.aspx

You'll add Custom Fields to the form that you name "Child 1", Child 2" etc
or whatever name you choose. I don't think you can add them to the Details
Tab so you might have to add tot eh General tab or use one of the other tabs
( (P.2, I.3 etc) and give it a name of your own.
It's really better if you follow a tutorial. It will be more descriptive
than me trying here.

Basically:
Open a Contact form
Tools | Forms | Design This Form
In the Field chooser click "New"
Give it a name and choose its type and format (text should be ok)
Then drag and drop the new field shown in "User defined fields in my folder"
to wherever you want.

repeat if you add other fields like child 2 etc.

Next you'll need to publish the form and update existing contacts to use it.
Info on that should be in the tutorial. The Outlook Forms group can give you
more info on customising Outlook forms.

It seems like a lot of work but lets you know the power of what you can do
yourself. I just did this with a new field "Child 1" while typing this.