Currently miners act as 'servers' for the Ethereum network, by doing all the needed work for Dapps to run, calculating, saving data etc..
Am I correct?
My question is this: who will run all of this after move to Proof of Stake?
Anyone can participate in block production by posting a bond. After posting a bond you have an opportunity to bet on which block will be included next. The incentives are such that you make money by betting with the eventual consensus and lose money by betting against the consensus. Any crypto-graphically provable misbehavior results in the forfeit of the bond.