What does "relayed connection" mean?
I've seen this when receiving files over skype, and some other situations, but I don't know what it means.
A relayed connection would refer to one that is being passed through one or more third parties, rather than being direct from the source to you. While your connection may be protected by encryption, it is possible for a third party to know that there is a conversation between you and another person and to see how much data is being transferred.
Skype uses some peer-to-peer networking features to build connections and network, this is one of those features that supposedly helps make the network more robust and reliable.
It means that a piece of software running on a machine independent from the two endpoints is tunneling network traffic between them.