Bitcoins are rare tokens, that can be exchanged in a provable way. There is no value promise whatsover, hence the markets for exchanging them. This whole thing works as long as the good part of the network has more cpu then the bad part.
Assuming that the overall amount of bitcoins comes closer to the 21 Million, and finding new coins becomes costly, who has an interest in keeping on using their cpus? There is some way of transaction fees, but if overall usage of the system would cease to grow, who would run it? It seems that the cpus need to run forever - if the good part of the network ever gets lazy, they whole system is lost?
To bootstrap the network, IRC is used, and also the other p2p clients. If the attacker controls my network connection, can't they just mimic a 'trusted' bitcoin network for the clients they control?
This seems to create a money transfer mechanism in which proof (signatures) replace trust. Accounts are used, but they are bound to public keys. It allows sub-accounts and creating of arbitary assets. Nice idea - an assets 'size' is described as a scale and a precision, with the numbers always stored as integers. (7,3) would turn 1234567890 into 123.4567890 and be displayed as 123.457.
Seems to be a nice system, but I don't see how it helps creating anonymous payments. It would rely on TOR etc, because the server can see the IP of the parties involved in a trade. Also having receipts is nice, but they only can become useful if they are usable in a dispute, e.g. when they are connected to a real world entity. But if that is the case, privacy is lost without even having a dispute.
Replaces the use of two books in double entry booking with the use of signed documents that describe the whole transaction, and are stored by all parties involved.