Short answers, pulled from the story.
Slush launched in November 2010 and became the first mining pool in history. This event occurred before individual miners faced an impossible reality where finding a block could take centuries if their processing power was too slow.
GHash.IO replaced Deepbit as the largest pool during the period between 2013 and 2014. F2Pool launched in May 2013 and eventually overtook GHash IO to become the new leader.
Mining pools use these shares to estimate exactly how much work each miner contributed toward finding a block. The probability of finding a block in any single share attempt depends on current block difficulty.
Pay-per-Share offers an instant guaranteed payout to miners for their contribution to the probability of finding a block. Proportional systems differ because miners earn shares until the pool finds a block which marks the end of the mining round.
Meni Rosenfeld invented the Geometric Method which ensures no advantage exists to mining early versus late in any given round. The method sets parameters so that scores granted for new shares remain constant relative to existing and future shares.