Biography
Our hero may be a perfectly Nice Guy, respectable, successful, a loving husband, and a good father. But what he really wants is for this one guy to acknowledge this.
Most often, that one guy is his emotionally distant father, though it can also be The Ace, The Mentor, an Aloof Big Brother, or especially that Always Someone Better individual, usually as an old friend of the hero. Almost always a Special Guest, and often a Large Ham as well. When it's a recurring character, usually a controlling parent.
A "Fawlty Towers" Plot will often ensue as our hero tries increasingly more convoluted schemes to convince the "Well Done, Son!" Guy of his worth.
In the end, one of two things happens. Rarely, the "Well Done, Son!" Guy will turn out to have some kind of massive character flaw, and our hero will realize that it's been a mistake to weigh his opinion so highly. This may result in the betrayed hero Calling the Old Man Out. Most of the time, though, everything comes to a head when the "Well Done, Son!" Guy finally asks the hero why he's gone to such ridiculous ends. The hero fesses up, and the "Well Done, Son!" Guy explains that he's respected the hero all along, and assumed the hero already knew. If the "Well Done, Son!" Guy is the hero's friend instead of his father, he'll often also explain that he'd always desperately wanted the respect of the hero as well (hell, sometimes the father wanted the kid's respect, especially if there's something big and nasty in his past, probably either ignored by or unknown to the hero).
Cue the Sentimental Music Cue.
In other genres, this can be a bit more understated, with the "Well Done, Son!" Guy simply giving our hero an approving nod from a distance (or saying, "That'll do, pig"). If the "Well Done, Son!" Guy is a Trickster Mentor or Zen Survivor, they may have a very long and painful road ahead of them to get even that. May be part of an Inadequate Inheritor plot. The Fantasy-Forbidding Father usually inspires this sort of feeling. This dynamic is usually father-son. Mother-daughter and father-daughter are not too unusual, but mother-son is rare (unless it's the Jewish Mother scenario, which is almost always Played for Laughs).
A really unpleasant variation is when the "Well Done, Son!" Guy is already dead, which in most cases means the approval and emotional bonding will never happen. At best, the character may gain some comfort by following his understanding of what the deceased would want if he were alive.
Contrast So Proud of You where the child receives their parent's approval. Also contrast Hates Their Parent where the child wouldn't be bothered with seeking their approval. For the inversion, see "Well Done, Dad!" Guy. This may be caused by Anti-Nepotism.