Utilities are structured differently depending on the state and region. I'll try to explain the basics:
Texas is fully deregulated meaning the utilities own absolutely no generation nor do they earn anything on electricity sales - they are only responsible for the transmission and distribution infrastructure. Independent power producers bid their energy into a wholesale market called ERCOT, and retail energy suppliers set the prices for customers. This creates competition at both ends and incentivizes the development of the best generation technologies. It just so happens that Texas has the most wind and fastest-growing market for solar.
At the other end of the spectrum are traditional regulated utilities that are monopolies. This is primarily the case in the Southeast and Western states aside from California. Electricity prices are determined by state public utility commissions, most if not all generation is owned by the utilities, and there is now alternative for the customers.
California is a rather unique case. They have a competitive wholesale market, but in many ways they are still a monopoly. However, municipalities can create their own utilities of sorts and procure energy separately. These are called community choice aggregations or CCAs for short. With the PG&E debacle, I expect this to take off and then the utilities would also just be responsible for the transmission and distribution infrastructure like in Texas. This is already about to happen for San Diego's utility, SDGE.
New England (ISONE) and New York (NYISO) also have their own wholesale markets, and customers have the option to buy energy from other companies. However, the utility remains the default supplier. Same is true in the Mid-Atlantic and parts of the Midwest (Ohio, Northern Illinois, and parts of Indiana) with PJM.
The rest of the Midwest (MISO), and areas surrounding Texas (SPP) have their own wholesale markets, as well, but the utilities are still the only option for customers.
Here's how the US electricity generation has changed over time. As you can see, the trends are pretty clear with coal dying, nuclear declining slower, while gas has taken off for now, and renewables are growing even faster. The Other is primarily hydro (using oil for generation is basically gone except for islands like Hawaii and Puerto Rico where it's also going away).
https://www.eia.gov/energyexplained/index.php?page=electricity_in_the_united_states. Again, the EIA is a very good source for historical data but its projections are questionable at best.