The day-star is the morning star, the herald of dawn, and literally a light-bringer. The expression may refer to the planet Venus or, more broadly, to the sun.
Isaiah’s day-star expression concerns the king of Babylon. In 2 Peter, the day-star is used as a title of the Lord and is associated with arising in believers’ hearts.
The King James Version renders the Isaiah passage “Lucifer, son of the morning,” while later renderings use “day-star.”