You can accomplish this by doing the following:
- Click your name in the top right corner and select Setup.
- Go to App Setup -> Customize, click on the object (eg Contacts), then Page Layouts.
- Click Edit next to the record type layout (eg Household Contact Layout).
- Find the field you want to be required on the sample record page.
- Hover over it and click the wrench icon.
- Check Required.
- Click Ok.
- Click Save.