The Martone case provides an example of a "mere expectancy" that is insufficient to make someone a "person interested," with legal standing to contest or establish a will. In that case, the decedent's will gave the executor of of his estate discretion to distribute some estate assets to his grandchildren, great-grandchildren, and other heirs. The executor did not have to distribute any assets to those heirs, but they could. Because those heirs had no concrete right to receive anything under the decedent's will, the Supreme Court said they were not "persons interested" in its probate.
On the other hand, a common example of a "person interested" in the probate of a decedent's will would be an heir who is excluded from the will, but would stand to inherit something under the laws of "intestate succession" if the decedent had no will. For instance, imagine a scenario in which a decedent is unmarried at the time of his death but has three children. If the decedent had no will, the three children would each be entitled to 1/3 of the decedent's estate. However, let's say the decedent signed a purported will that left everything to one of the three children, but nothing to the other two. If the two children excluded from the will had grounds to believe it was invalid (for example, if the parent lacked the mental capacity to sign it or because it was procured by undue influence), they would certainly be "persons interested," with legal standing to contest the will.