По-английски means "the English way", по-спортивному means "the sport way/style" по-дружески means "the friendly way/style", по-военному means "the military way/style", по-братски means "the brotherly way" etc.
So "Ты хорошо понимаешь по-английски?" literally means "Do you understand [something said/wrttten] the English way well?" while "Ты хорошо понимаешь английский?" means "Do you understand English [language] well?"
По-английски can mean other things done in English way as well "он ушёл по-английски" literally means "he went away in English style" but ideomatically it means "he went away without saying goodbye". Мясо по-французски means "meat prepared in French style" etc.