DID is not a VoIP term and is not a geek term; it ought to be known to most people contactable over the telephone, in their office, in medium to large companies in countries that provide it.
DID means the ability to dial down to a particular extension on a PABX from any public network telephone. E.g., looking at Digium’s contact information, they almost certainly have DID with all numbers of the form +1 256 428 6xxx being routed to their PABX.
Peer to peer connection is good for connected calls, where the network allows it. In the Skype sense, it is bad for call setup or when the IP network doesn’t allow direct connections, because it relies on infrastructure borrowed from other users and not under the direct management of the network owners.