AWS allow you to create your own private network 'in the cloud'. They call this VPC and it's fairly east to setup.
You can connect to your VPC in a number of ways, two examples:
You can connect to instances in your VPC via VPN (over an Internet Connection).
You could also use Direct Connect to connect to instances in your VPC via a private network (ie NOT over the Internet).
- Better ping/response times (more reliable)
- More secure (as it bypasses any and all Internet traffic)
- More costly (there are setup/ongoing charges with Direct Connect)
VPC & Peering
Also have a look at peering for communication between VPCs.
Connect privately to other VPCs- Peer VPCs together to share resources
across multiple virtual networks owned by your or other AWS accounts.
Direct Connect & VPN
Setting Up AWS Direct Connect with VPN http://docs.aws.amazon.com/govcloud-us/latest/UserGuide/setting-up-direct-connect-with-vpn.html