In reply to fmck:
Its it a Kindle Fire Kids Edition, or just a standard Fire? With the Kids Edition they are preset so kids can't purchase stuff or use in app purchases without a passwords.
There is a similar feature on the normal Fire tablets (Look for a 'Fire for Kids' logo) which runs the kindle in restricted mode and lets you determine what apps/games they have, and prevents internet access. A password is required to exit the Fire for Kids mode, so they can't bypass the settings.
If you don't want to do the above, you can leave the the fire in its 'regular' mode, and switch on the 'parental controls' (go to settings>parental controls, enter a password), which again prevents them purchasing stuff and using the internet.
You may have to restrict the 'in app' purchasing separately. Go to Appstore>settings, and click to restrict the 'in app purchasing'.
Check the amazon help pages, it does tell you how to do this, but can be a little hard to find.