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.