Definition of profiles in Microsoft words:

Domain. Windows automatically identifies networks on which it can authenticate access to the domain controller for the domain to which the computer is joined in this category. No other networks can be placed in this category.

Public. Other than domain networks, all networks are initially categorized as public. Networks that represent direct connections to the Internet or are in public places, such as airports and coffee shops should be left public.

Private. A network will only be categorized as private if a user or application identifies the network as private. Only networks located behind a NAT device (preferably a hardware firewall) should be identified as private networks. Users will likely want to identify home or small business networks as private.