B-good points on insurers, but you left out another big group-reinsurers. Even Geico, etc may not actually be your "insurer". All insurance companies use reinsurance to further protect them from policy losses. Munich Re and Swiss RE being the largest in the world, each with about $25 Billion in premium income. It is sort of like a local bookie who takes bets larger than he can handle and them "lays" them off to a larger bookie.
On insurance pools-years ago I spent some time with an actuary who spent several days trying to explain insurance to me, it was for a paper I was doing for an economics class. It was pretty fascinating. It boils down to two pretty simple concepts (1) how you set up and define a risk pool, and (2) how detailed information can you gather about the risk pool. The smaller and ore homogenous your pool, the easier to detail your prospective payouts and to price your premiums accordingly.