When considering the most eco-friendly way to wash your dishes there are a few things to consider:  water use, energy use, and chemicals. 

Water Use

The average dishwasher  uses 6 gallons of water per cycle, while the average Energy Star-rated dishwasher uses 4 gallons per cycle.  Older dishwasher manufactured before 1994 can use between 8 and 15 gallons per cycle.  Pre-rinsing your dishes before you put them in the dishwasher will add to the water usage.  The average faucet flows at 2 gallons per minute.  So to be using water at the same rate you would need to wash your dishes with about 2 minutes of flowing water time.  An Average dishwasher holds about 8 place settings (plates, bowls, forks, knives, spoons, glasses, etc.), so if you can wash all of those dishes with only 2 minutes of running water, that's 4.4 seconds of flowing water per piece, then hand washing and could be a competitive option.

Energy

The energy use used in washing dishes mostly comes from heating the water.  Dishwashers, on average, use between 0.87 kWh of energy per load to 1.59 kWh of energy per load.  Dishwashers have the capability to heat only the water used for the dishwasher and not all of the household water.  Dishwashers will also use some energy if you dry them in the dishwasher as well.  Heating the water in your faucet takes a variable amount of energy depending on the amount of water being heated and the type of water heater used.  Instant water heaters are the most efficient when compared to gas storage tank water heaters and electric tank storage heaters.  A study done by the University of Bonn in Germany observed that it took on average 2.5 kWh of energy to hand wash 12 place settings. 

Chemicals In Detergents

Machine-dishwashing detergents often contain phosphates.  Phosphates are naturally occurring substances, but if too much of it gets into waterways, algae and phytoplankton feed on it and reproduce in massive numbers; causing an algal bloom.  Algal blooms block out sunlight and cause oxygen imbalances in a body of water and as a result kill off other species living in the water.  Phosphates often times cannot be removed by wastewater treatment plants.  The dish liquids used for hand washing dishes often do not contain phosphates.   However, dish washing liquids can contain numerous other chemicals like chlorine, alkyl phenoxy ethanols (APEs), dichloromethane (DCM), diethanolamine (DEA), dioxane, phosphates, sodium lauryl/laureth sulfate, and synthetic fragrances that can contaminate the water supply when washed down the drain.

So Which Is Best?

If ran when at capacity the best choice for washing dishes an Energy Star rated dishwasher with a no or low phosphate dish detergent.  An added bonus for using a dishwasher is that it saves you time.  It can be possible to hand wash dishes with the same energy and water efficiency, but you have to be very conscious of your water use. 

If your looking for an Energy Star rated dishwasher check out these great options: