For technical specs, I would suggest you to read The Paper: http://primecoin.org/static/primecoin-paper.pdf
For the know-how of network and transactions, you can read some well-written articles on Bitcoin, as PrimeCoin differs mostly in Proof of Work algorithm. (we find prime chains instead of computing SHA256)

From a miner’s point of view, prime numbers (more specifically: prime chains) are used as a “proof of work”, because they are hard to find (especially with given constraints).
On the other hand, these constraints are derived from a sum of previous transactions, which are “confirmed” by the mined block.

The coin itself doesn’t exists, your amount of money is determinated by a shared history file of transactions. Your balance is so determinated by a signed sum of transaction that contains your address

ie.
mine -> A 100 XPM
A -> B 30 XPM
B -> C 2 XPM
A -> D 4 XPM

in the end if those transitions are known we have that
A has 66 XPM
B has 28 XPM
C has 2 XPM
D has 4 XPM