lungs; bottom of one's heart; vital point
Literally 'lungs', but more often used figuratively in expressions like 肺腑をえぐる (to pierce one's heart) or 肺腑にしみる (to sink deep into one's heart). The figurative sense emphasizes deep emotional impact or the core of one's being.
His words pierced my heart.
The beautiful melody sank deep into my heart.