When something is done "by hand", it means that a person does it instead of a machine. In the example above, washing dishes "by hand" means washing them in the sink instead of in the dishwasher. It can also mean doing something yourself that people usually hire professionals to do
Other things that you can do "by hand" include:
build a house by hand
wash your clothes by hand
mix the dough by hand
score a test by hand
count something by hand
Notice that it's always "by hand", not "by my hand", "by her hand", "by hands", etc.