I know that the US has three or four major electricity networks (East, West, Texas, Hawaii) but I dont understand how they are they are regulated or operated.

In many countries there are generators who produce power, retailers who sell power to retail customers and network operators who 'move' power between generators and consumers either through high voltage or local transmission lines but these roles are separate and you pay a separate fee for the connection/transmission vs the power you buy. Retailers pay to 'move' power from where its produced to where their customers are.

The transmission companies in most cases regulated natural monopolies. Retailers and producers can be the same company.

How does it work in the United States? Does one company own everything in some areas? Do you usually have a choice of energy retailer?