When you create a new contact for an organization on CiviCRM, there does not seem to be any place to add the name of a human contact. Am I missing something?
Both organizations and individuals are 'contact' entities in CiviCRM (a household is also a 'contact'). An organization is a separate 'contact' from the individuals related to that organization. By having the organization entity separate from the people in/related to that organization, you can specify the nature of the relationships those people have to the org.
You'll want to create an organization (Top menu > Contacts > New Organization, as you have done) then separately create individuals (Top menu > Contacts > New Individual) who have relationships to that organization. You can connect the individual(s) to the org with either the Current Employer field or using the relationships tab on a contact's summary screen once a contact has been created.
You can read more about contacts and relationships here: http://docs.civicrm.org/user/en/4.6/organising-your-data/contacts/
You don't. a human contact is a contact type "individual", what you create is an organisation.
If you want to create a human, create an individual, and set the current employer as the organisation. it will automatically create 2 contacts, the individual and the organisation, and create a employee/employer between them
you can as well later on update the relationships between organisations and individuals from the relationship tab. You can even create new type of relationships (volunteer, member of the board...) on the top of the standard employee.
I also often like to have an individual associated with an organisation, but don't want to create a separate record for them, although that is clearly the more flexible option.
So I have created a simple Custom Data field that I call "Main Contact" that appears on the Summary page of an Organisation contact type, with a simple text field for the main person associated with the organisation. It's not hard to do and fits our use case much better than creating a whole new record for that person. It also means you can customise the Email or Postal Greeting to have something like "Dear Mr Smith, Duke Street Church".