A key part of the new Azure is Virtual Networks. This features lets you connect a network of Azure VMs to your local network over a gateway virtual private network (VPN). Currently this requires you to use one of a limited range of VPN security devices from Cisco or Juniper, with other hardware possible but not supported.
The benefit is that VMs on Azure become a seamless extension of your on-premise network. You can join Azure VMs to your Windows domain, or host a domain controller in Azure. There is no need to think about how to synchronise local users and Azure users since they all use the same Active Directory. Since Azure VMs can be deployed or removed on demand, one benefit is that you can use Azure to scale out your IT infrastructure without buying new hardware.
Azure also offers a machine-to-machine VPN connection called Azure Connect.