Hi Big heart,
both , liǎng gè dōu / liǎng zhě dōu , 两个都 / 两者都
each , měi gè , 每个
each other , hù xiāng , 互相
either , yě / rèn yì yī gè , 也 / 任意一个
everybody , měi gè rén , 每个人
everyone , měi gè rén / měi yī gè , 每个人 / 每一个
everything , měi jiàn shì , 每件事
I hope my answers can be useful for you to understand these phrases. If you have any problems about Chinese, just let me know.
^_^ Anna