to bow down before someone in reverent homage: to worship
The picture behind this word is physical: a person lowers the body, kneels, or falls face-down before another to show that the other is higher. The same gesture could be offered to a king, to a powerful person whose favor you needed, or to God, so the word covers a wide range, from a desperate request made at someone's feet all the way up to full worship of the living God. The body language is the constant; who receives it is what changes.
In the New Testament that range narrows around one urgent question: who is worthy to receive it. Toward God and the Lamb the gesture is right and commanded. Toward false gods, idols, the beast of Revelation, or Satan it is the central sin, devotion stolen from the one to whom it belongs. And when people fall at the feet of Jesus in the Gospels, the gesture sits exactly on the seam between deep respect and true worship, which is part of why the Gospel writers use the word so deliberately.
The senses below sort the occurrences by who is on the receiving end, because that is what decides whether the act is faithful, sinful, or simply the courtesy of bowing before a person.