Which of the following best defines one mole of a substance?

Mass, in grams, of each particle of the substance
Mass, in grams, of the nucleus of the substance's atoms
Amount of substance present in one atom of the substance
Amount of substance present in Avogadro's number of particles of the substance

1 mole = 6.022x10²³ of anything.  The number 6.022x10²³ is known as Avogadro's number.  Avogadro's number is not a mass, it is just a unit of measure.  Other units of measure include: 1 pair (2), 1 dozen (12), a baker's dozen (13), et. al.  1 mole is similar in that it simply represents a certain number of things; specifically, Avogadro's number of things.

Since we know 1 mole is not a mass, the first two options can be eliminated.  On observing the third option, it cannot be correct because, as we said, 1 mole is equal to Avogadro's number—not an atom of a given substance.

Amount of substance present in Avogadro's number of particles of the substance

the answers would be

the first one and the second one