There are two main VoIP camps – H.323 (grandaddy) and SIP (newcomer).
H.323 has a much richer featureset, but setting up and tearing down a call is complex.
SIP is a simpler featureset, and simpler to manage. Vonage uses SIP.
The gateway server connects your box to the phone network, and vice versa.
SIP Vonage Phone
Box Gateway System
SIP connects to the server and leaves a connection open. This is how the Vonage gateway server knows how to find your Vonage box, no matter where it is. When someone calls you, they actually connect to the Vonage gateway server, which then does its VoIP thing, and sends the call to your box.
The gateway server is hardcoded into the Cisco box, so the box always knows how to find the Vonage gateway server. When you dial out, the phone number is sent to the gateway server, which then does its VoIP thing again (in reverse) and connects your voice stream to the actual phone network.
Hope that helps a little?