There are 2 different meanings of P2P, and you seem to be getting confused.
P2P can mean Point to Point - as is the case with a VPN client like OpenVPN or PPTP. These connections occur directly - and only between 2 devices.
It can also mean Peer to Peer, as in the case of bit torrent etc, where there is a web of connections. The rest of this post deals with these cases, as they seem to be what is of interest to you.
There is no rule for how a P2P networks can be made - it depends on the protocol. That said, there are usually a number of servers and a number of clients.
The servers role is typically to help the clients find each other.
Once the clients have found each other, they normally communicate directly - traffic between the clients DOES NOT pass through the server (and indeed, if it did it would not be a P2P network).
Depending on the type of traffic though, there does not need to be a server or list of servers - in limited circumstances its possible for all clients to advertise their presence - although this is not common across the wider Internet as it does not scale well.
An IRC server is a simple, old-style chat server - it is possible for a P2P network to use this to co-ordinate communication between clients, but this is unnecessary, and not common. (IRC servers are often used to control botnets while hiding the controllers details). A tracker is a more common mechanism for handling this function.
I'm not an expert on Bitcoin, but I understand that no tracker/central point is required - indeed one of the advantages of it is totally decentralised (ie does not require a central point or tracker) - and that each bitcoin has a chain (ie can be tracked back to its original creation).
A Bitcoin tracker is not actually required for Bitcoin, and is, AFAIK a way of finding out the value of bitcoin based on trades being made.
