Cryptocurrency networks (such as Bitcoin, Dogecoin) use what's known as a block chain to store the history of all the transactions ever made. Using this history, we are able to figure out the balance of each and every account - this is how the network knows how much money is stored in each address. As you'd expect, bits of data known as blocks make up the block chain. Each block contains a list of transactions made after the previous block.
Mining is the process of creating these blocks. In order to regulate block creation, the network demands that the block must contain the solution to a computational problem with a set difficulty. As a reward for solving this problem, the miner is given coins. The difficulty of the problem is adjusted based on how long it took for the last problem to be solved.
As the amount of miners on the network grows, it is becomes non-viable for a miner working alone to solve the problem in a reasonable amount of time. Pooled mining is a service that that allows many miners to work together and split the rewards.
The P2Pool software encodes the shares into what's called the share chain. This is very similar to the block chain used by Bitcoin. This chain is built using the computational power miners use to find shares. All of the P2Pool nodes maintain a copy of the share chain for the current block and use it to determine payouts to miners.
Due to the way the PPLN (Pay Per Last N Shares) payout system in P2Pool works, slow or very unlucky miners may not receive the first payout for up to 24 hours.
However, once the first payout is received, the payout for the miner will grow to match the miner's mining speed. The system is designed to punish pool hoppers - miners who switch pools to get the maximum amount of shares.