In order to establish the connectivity between two nodes, the first thing we need to do is actually discover the peer to connect to. Traditional solutions like TURN/STUN/ICE does not address this. Below are the commonly-used techniques to discover peers.
This is how airplay/DLNA etc discover your other devices. Unfortunately, mDNS relies on multicast to work, which means normally it works only on your LAN (not always, for example, mDNS zerotier works over WAN by creating virtual network devices).
This is how bittorrent etc discover which nodes to fetch the data. One downside of DHT is that, unlike mDNS, it requires bootstrapping nodes. It is normally expected that the network nodes have publicly accessible network address.
This is how nodes in bitcoin network etc broadcast out the new mined block. One of the greatest advantage of gossip protocols are that participants do not need public IPs. What they need to do is to keep connection to nodes that do have public IPs. Gossip protocols are susceptible to the same weakness of DHT, i.e. requiring bootstrapping and are quite verbose.
Having learned the identity and addresses information of the peer to connect, we can now try to establish connection between peers. Depending on the concrete network situation, there may be a few possibilities.
In case when the peer do have a public IP and the gateway router supports active port mapping standards like UPNP-IGD or NAT-PMP, all we have to do is ask the route to actively forward all traffic to some ports on its WAN interface to our program.
If we are not so fortunate to have such a luxury, we then need the coorperation from the peer to punch a hole in the NAT. In order to do such thing, we need to first notify the peer our intention. That is why a both connected-to signalling server is needed. Libp2p makes this signalling process more decentralized by first everyone can be a signalling server and second the association of one peer to the signalling server can be learned decentralizedly (by DHT or gossip like process).
As a last resort, we can relay traffic through an inter-mediate server. This is not scalable.
After we have create a stream (basically a normal network connection, but multiple logic stream can be multiplexed on a single network connection), we can then forward traffic between local and remote socket. This is the least intriguing part. We only want to emphasize that a signle established connection can be multiplexed to several streams. This way, we can create many forwarding pipes as we wish.