Azure is an Infrastructure and Platform services offering. Azure exposes some of its services like Storage, SQL, and Websites over a Public peering, while services built and connected on the Virtual network are exposed through a private peering connection.
Customers can connect to all public peering services through the Internet or through Express Route. The virtual network created in Azure can connect to the on-prem network through VPN. Virtual machines exposed over terminal services are accessible over internet public IP address.
In summary, Azure exposes the Internet and private connectivity options to customers.