What is the meaning of the words "certainty" (al-yaqīn), "the eye of certainty" (ayn al-yaqīn), and "the light of certainty" (nūr al-yaqīn)?

Certainty (Al-Yaqeen) is knowledge and the removal of doubt.

"The eye of certainty" (Ain Al-Yaqeen) is direct observation and witnessing, as in the Almighty's saying: "{You will surely see Hellfire. Then you will surely see it with the eye of certainty}" (102:6-7), and this is the highest rank of certainty.

"The light of certainty" (Noor Al-Yaqeen) is the expansion of the chest for faith. It is a light that Allah casts into the heart after striving, causing the heart to be filled with light and the chests to become radiant. This is what is mentioned in the Almighty's saying: "{Is he whose heart Allah has expanded for Islam, so he is upon a light from his Lord?}" (39:22). So, whoever's heart is illuminated by the light of certainty, his soul perceives the good consequences and his desires die, whereas whoever's heart is overwhelmed by the darkness of heedlessness is blind of heart.