Efficient dishwashers often use less water than washing by hand. According to Green Choices, dishwashers use three to four times less water than hand washing when cleaning a twelve-place table setting.
In terms of carbon footprint, The Guardian quotes these figures by Mike Berners-Lee:
Almost zero CO2e: by hand in cold water (but the plates aren't clean)
540g CO2e: by hand, using water sparingly and not too hot
770g CO2e: in a dishwasher at 55°C
990g CO2e: in a dishwasher at 65°C
8000g CO2e: by hand, with extravagant use of water
It is important to note, as Berners-Lee and Clark do, that the figures above also include the energy required to produce the dishwashers themselves and assume you keep a dishwasher for 10 years. As these figures show, if dishes are to be properly cleaned, then using the dishwasher is the best option. The least wasteful hot-water hand-washing can beat a fully loaded dishwasher in the UK when an efficient, modern, gas-fired boiler is used, but few people are really careful about wasting hot water when washing up.
A different study at the University of Bonn (cited by Larry West) also favoured dishwashers, finding that dishwashers used half the energy and one-sixth of the water used in hand washing identical sets of dishes.
If hand washing is the only option available to you, then try to create a lather of washing up liquid before rinsing items sparingly, instead of filling up a bowl with hot water that goes dirty and has to be refilled.
If you do have a dishwasher, make sure to run it on an eco setting (if it has one) or lower temperature, scrape - but don't rinse - plates beforehand, only put it on when it is completely full, and try timing it to run at night when energy demand is at its lowest. Avoid using features like rinse-hold, pre-rinse, dry and heat-dry - the dishes will dry quickly if the door is left open after the wash and rinse cycles are complete.
Berners-Lee, M., and Clark D., 'What's the carbon footprint of... doing the dishes?', The Guardian (August 2010), accessed at: https://www.theguardian.com/environment/green-living-blog/2010/aug/19/carbon-footprints-dishwasher-washing-up
Green Choices, 'Is it greener to hand wash or use a dishwasher?' (2020), accessed at: https://www.greenchoices.org/green-living/at-home/is-it-greener-to-hand-wash-or-use-a-dishwasher
West, L., 'Eco-Friendly Kitchen: Dishwasher or Hand Washing?', ThoughtCo. (2 March 2018), accessed at: https://www.thoughtco.com/are-dishwashers-good-for-environment-1203932