Crouching Tiger, Hidden Dragon: Subverted. The apprentice (Jen) is a more skilled combatant than the Master (Jade Fox), but the Master is more treacherous.
Star Wars is the Trope Namer. The Sith have the Rule of Two, where there can only be a Sith master and an apprentice - no more, no less - one to possess power and the other to desire it. The mentality is that each Sith Lord teaches one apprentice until they are strong enough to destroy the master and take their place, at which point they will seek out a student of their own. In this manner, each master not only becomes strong, but must stay that way to stay alive, and each student must become stronger than their master. Each generation of Sith is stronger than the last, because any weak link in the chain is dead. A master cannot have more than one apprentice, because they may team up to kill their master despite being individually weaker than him, then turn on each other and weaken the Sith Order. In short, Chronic Backstabbing Disorder as a way of life and a code of morality.
The origins of this rule are explained in Darth Bane: The Path of Destruction, where the Sith have become rife with in-fighting due to hordes of individuals claiming the title of "Dark Lord of the Sith" with a leader pretending to be just a first among equals. The Sith philosophy in such a climate can only lead to self-destructive power struggles that serve only to empower their enemies. Bane eventually comes to the conclusion that there should only ever be two Sith Lords at a time: one to hold power, and the other to covet it. Eventually the apprentice should take it by force when they're ready. Furthermore he concludes the Sith should not work for the fall of the Jedi and to conquer the galaxy by armed conflict, but rather subtlety, which Palpatine eventually pulled off. Having made up his mind on this, Darth Bane makes sure all the collectivist Sith Lords unintentionally commit collective suicide.
Interestingly, the Expanded Universe reveals that Palpatine, Dooku, Vader and other Sith Lords had their own minions trained in the dark side; the rule wasn't technically broken because they weren't "true" Sith Lords, but mere warriors or spies with some lightsaber training and ability in the force. However, it seems that once a minion begins to grow too strong, or - worse - is discovered being trained in the Sith arts, one of them has to die, be it minion or master. Or potentially the apprentice, with the minion becoming the new apprentice.
Palpatine, for example, deceived his own master into thinking that Darth Maul was simply an assassin, not a true apprentice. The one thing that didn't go according to plan was that Maul was killed right after Palpatine disposed of his own master...though Palpatine made that work to his advantage by recruiting Dooku, who was better suited than Maul had been for the very public role needed in the next phase of the plan.
Darth Tenebrous broke the rule by recruiting two apprentices and not telling them about each other. After Darth Plagueis "took over", he was surprised by Darth Venamis, who had learned of him, was better at the use of the lightsaber and had been trained in Plagueis's style. Plagueis was smarter, imprisoning Venamis and experimenting on him for years. It's possible that Venamis wasn't a "true Sith" though.
Ultimately Darth Bane's philosophy also resulted in the Sith's undoing, as Bane did not take into account the possibility that either the master or the apprentice could actually be redeemed, as evidenced by Darth Vader's Redemption Equals Death and Jacen Solo/Darth Caedus' apprentice renouncing the Dark Side of the Force and the Sith shortly after the former's demise. What's worse is that Bane really should have known better. The Sith Lord whose Holocron gave Bane the inspiration for the Rule of Two, Darth Revan, had also renounced the Dark Side, and Bane knew this.
Live Action Television
In Heroes the Company's rule is "one of them, one of us".
Characters fight in pair during Imperishable Night.
Yukari Yakumo's ability is explicitly defined as the ability to manipulate border between two different things. As for what that means in practice, it apparently means whatever Yukari wants it to mean.
For that matter, the yin-yang features heavily in Touhou; it's even one of Reimu's attack mode.
Prominent in Fate/stay night. Sure, you could just kill the Master and run away from the Servant (it's even repeatedly stated to be the best strategy), but nobody ever does. Most apparent with Kotomine/Gilgamesh and Kuzuki/Caster. Nobody in either pairing is really the boss. Gilgamesh will do what Kotomine says, if he feels like it. Caster would obey Kuzuki's orders, but Kuzuki is too passive to really bother most of the time. Also apparent in Kotomine's fight against True Assassin and Zouken Matou. He's stronger than Assassin, but can't kill him due to him being a Servant, so he has to take out Zouken first and then Assassin would be vulnerable. But Zouken is essentially unkillable by normal means plus Assassin is keeping him busy.
In Evil Plan. Dr. Kinesis hires Alice as his second in command because he loses his previous number two.
This is a common theme in Kim Possible. Drakken and Shego, the Seniors, and even Gemini says that he was saving the "Alpha" title for someone special.