In PPS or RPPS pools miners are paid for every valid share they submit, generally using the formula: (Block reward - pool fee)/difficulty. In this system the pool owner takes all the risk in terms of the amount of time/blocks it takes to solve a block as the miners get paid regardless.
The longer it takes to solve a block, the more shares the owner has to pay out of the block reward when a block is solved. If it takes too many shares the owner will take a loss on the block.
On the other hand, if blocks are solved quickly, the owner makes more profit. The miner is insulated as they get paid a consistent rate regardless of the "luck" the pool is having.
The formula is applied to every block change, regardless if the pool solves the block. When the difficulty is low shares are worth more and thus the owner pays more to solve a block. Normally not a concern as the lower the difficulty the speed and amount of blocks that are solved rises.
When the difficulty is fluctuating wildly, it puts the pool owner in a extremely uncertain position. When the difficulty is being purposely manipulated, all consistency goes out the window for the operator and even when the difficulty is low, the pool is still in competition with somebody with enough hash power to manipulate the chain in the first place.
In proportional pools, the risk is taken by the miners. The basic formula is (block reward - pool fee)/ number of shares to solve the block. The longer it takes to solve a block, the less each share is worth. The pool operator never has to pay out more then the block reward - pool fee. In reality proportion pools use a slightly more complex formula to discourage pool hopping, but the basic premise is the same.The only real risk to the owner is making enough to cover hosting/development costs, as do PPS pool owners. Most proportional pools run at a very low fee or even no fee, only making coin on the block fees.
Most pools today are PPS, and as a miner I will normally only use PPS pools.
RPPS/RBPPS are round based PPS pools that delay payout until the solved block matures. This puts the risk on the miners for orphan blocks; it is also how young pools can start up with fewer coins in reserve.