The author penned the poem of the song, sepcifially referrign to the Grace of God. The author wrote, while at sea that "only the grace of god" could have saved him.
It is based on 1st chronicales chapter 17, versus 16 and 17. (which does not mention the word grace at all).
The verses are a statement about Devine Grace which is in christiandom: sovereign favour of God for humankind.
It embodies Gods gifts to man kind, and God's power.
Amazing Grace, in the song "Amazing Grace" is talking about god's love and power to heal. It means that God stands by our side every day, in every hour, and helps us, and guides us through our troubles.