Once upon a time, we washed everything in soap, a natural surfactant, or sudsing agent, made from fat (animal or vegetable) and lye. Soap is fairly effective in soft water. But it doesn't cut grease well in hard water, and it tends to leave a soapy film on things.
Nowadays, we use detergents. They work a lot better.
The surfactant in detergent is made from petrochemicals. Most detergents for washing dishes by hand also contain a softener, a fragrance and a colorant.
They do not contain phosphates, bleach or ammonia. And they are fairly mild, as detergents go.
Automatic dish-washing detergents are a different story. They can't rely on surfactants to get dishes clean, because suds escape the machine and fill the kitchen. And they can't rely on mechanical action to wipe dirt off dishes, since there are no busy little hands inside your machine. So automatic dish-washing detergents tend to be much stronger than their hand-washing equivalents.
They also contain a lot of phosphates, even in states that have banned phosphates in laundry detergents. Phosphates are a problem because they feed excessive algae growth in fresh water, choking lakes and streams with masses of algae.
Phosphates have an associate an element that tends to accompany them as they are mined from the Earth, added to your detergent, poured into the machine, down the drain and into wastewater. That associate is arsenic, and where it ends up is in sewage sludge, the solid material recovered from wastewater.
Arsenic from dish-washing detergent isn't a huge problem. But it does contribute to a much bigger problem many communities have: Their sludge is contaminated with toxic heavy metals so cannot be used as fertilizer.
All major brands of automatic dish-washing detergent liquid and powder also contain chlorine bleach. They will react with ammonia to release dangerous chloramine gas. Because of this, you should NEVER use automatic dish-washing detergent to wash dishes by hand.
Fortunately three small companies now make automatic dishwasher detergents that are largely free of phosphates and chlorine. These are Mountain Fresh Products, with Kleer II and Kleer III gels, Bi-O-Kleen and Life Tree. These may not be on your grocery store shelves, but you can find them at most natural food stores and coops.