Biography
This trope is about two individuals (or occasionally, factions or civilizations) that are significantly linked together somehow (such as through a rivalry, friendship, being siblings, co-workers, competitors, etc.) that have differing personalities and, with it, often different approaches to the world in general, especially whatever they might be competing in/for/over. This, of course, gives them something to butt heads about, since their manner of acting may rub each other the wrong way and each is sure that their way of doing things is right.
By the way, don't get too stuck on the colors in the title. The title refers to a Japanese tradition that gets elaborated a bit later in the article, but it doesn't mean that a character has to dress in or be associated with a particular color in order to fit the role.
Their personalities usually run roughly along the following lines:
The Red Oni is associated with passion, wildness, and defiance. A Red Oni character is often more brawny than brainy, extroverted, enthusiastic, determined, and filled with a zest for life. They are also much more likely to break conventions and rules than their counterpart. Often an Idiot Hero or, in more mature pieces, a Boisterous Bruiser. If a Red Oni is indeed noticeably intelligent, they will still be much more wild and impetuous than the Blue Oni and/or with less sophisticated thoughts, feelings, or goals. Generally speaking, they're more people-oriented than goal-oriented (although both Oni's can be either). They may or may not shrug off criticism about themselves or their aims and goals, but say or do something nasty to their closest friends and you've just guaranteed yourself a world of pain. When a goal-oriented villain, the Red Oni will often be Ax-Crazy and psychotically willing to sacrifice any innocent lives to achieve their goals. A more Affably Evil Red Oni tends to be a Blood Knight. In terms of character alignment, the Red Oni tends towards the Chaotic alignments, though this isn't always the case.
Freudian and Power Trios: The Id, may be The McCoy (though not guaranteed)
Four-Philosophy Ensemble: Optimist or Apathetic
Four-Temperament Ensemble: Sanguine or Choleric
Five-Man Band: The Leader/The Hero, The Lancer, or The Big Guy
Dere types: Deredere, Tsundere, Yandere
The Blue Oni is associated with serenity, control and observing authority. A Blue Oni is more intellectual, proud, traditional, introverted, and cultured (sometimes more spiritual, although that's not guaranteed). Blue Oni personalities are often respected by others, but also likely to puzzle or confound their peers because they are difficult to read and have a mysterious quality to them. Personality subtypes include The Stoic, The Quiet One, Aloof Big Brother and Deadpan Snarker. It's worth noting that while blue types appear tremendously calm and composed, sometimes, this is just the surface, and scratching that façade may show that they are, if anything, wound far too tight beneath their calm and unemotional appearance. Villainous Blue Oni tend to be willing to sacrifice anything (and often anyone) for their goals, mostly only when their villainous goals are laden with true purpose (see Knight Templar and Well-Intentioned Extremist). Being more calculating, conniving and thorough (and if The Mole comes into play), Blue Oni tend to be more successful villains long-term than Red Oni. In terms of character alignment, the Blue Oni tends towards the Lawful alignments though this isn't always the case.
Freudian and Power Trios: The Superego or The Ego, may be The Spock or The Kirk (though not guaranteed)
Four-Philosophy Ensemble: Cynic or Realist
Four-Temperament Ensemble: Melancholic or Phlegmatic
Five-Man Band: The Leader, The Lancer (If the leader/hero is a Red Oni), The Heart, or The Smart Guy
Dere types: Kuudere, Dandere, Himedere
While many expressions of this trope play it straight, it is also not uncommon for the Blue Oni and Red Oni to mix traits. The Red Oni, while usually the more passionate and emotion-driven, may also be the more easy-going and flexible of the two. Their more simplistic world-view may make them at least partially immune to angst and Wangst. Conversely, the Blue Oni, while usually stoic and thoughtful, may hide a volatile, angsty nature that may erupt if pushed too far. In such instances the two may switch roles, with the Red Oni becoming the reassuring, composed partner and the Blue Oni becoming a Whirling Dervish of Rage! [Compare pairings such as Red Oni Ken (cocky, arrogant, but decidedly easy-going) and Blue Oni Ryu (stoic, disciplined, but brimming with internal conflict)].
If Elemental Powers are present, you can usually expect the red oni to use Playing with Fire or something of a similarly flashy nature, while the blue oni will prefer Making a Splash or possibly be An Ice Person and an Elemental Rivalry is born.
There are a lot of ways to play their interactions, so sometimes the two are both leads on an equal standing, sometimes the blue is more of The Smart Guy or The Lancer to the red's Hot Blooded The Leader or The Big Guy. The relationship can also vary wildly, as they are just as likely to be close friends as they are to be bitter rivals or enemies. One thing that is sure is that regardless of how big a cast is, when these two are in it, they will be magnetically drawn to each other.
Note that many times, this will be a relative, not an absolute, distinction: the Red Oni need not be The Fool and the Blue Oni need not be a Straw Vulcan.
The name comes from traditional Japanese folklore where Oni (think troll/ogre) usually came in two flavors, a Red and a Blue. Over time, it's gotten popular to ascribe different qualities and personalities to the two colors. (Think the high-strung Type A personality and the easy-going Type B personality.) See also the Marid and Ifrits over at Our Genies Are Different.
In a show where it's relevant, differences in fighting style may be something like Technician Versus Performer. For example, a red may rush headlong into combat without focus or a plan, relying on surprise or intimidation. It may be that their strategy is to close the fight quickly before their opponent can think or adapt. It's also possible that they're going to tank their opponent's defenses in order to focus on beating them down. The blue may use cunning traps, stealth and strategies, or tend to win in elegant Single-Stroke Battle fashion, or rely on hitting a target with magical powers or projectiles from a distance instead of running up to their opponent and punching them in the face. Force and Finesse is a variation where red tends to rely on raw power while blue uses precision strikes. Analysis page has more examples of possible red-blue dynamics.
Male-male oni pairs in Japanese-language works usually have an Expository Pronoun contrast, most often with the red oni calling himself ore and the blue oni favoring boku.
Add one person (We'll call them the Purple Oni) that bridges the Red Oni and Blue Oni and you got Freudian Trio, with the Red Oni becomes the Id, the Blue Oni becomes Superego, and the Purple Oni becomes the Ego (or a Blue Oni is the Ego as well, whose either torn between the two, or not involved.). That's the basic formula, but not always. Sometimes as the plot progresses, someone new is added and as they become the trio, this newcomer might be even more emotional than the Red Oni or is even more calm and calculating than the Blue Oni. And whoever keeps the trio together is the ego.
Keep in mind that this trope can apply to anyone and to any pairs of people. They do not have to literally wear red and blue for this trope to work and sometimes, they can even wear opposite colors that contrast their roles as a red or blue Oni. They also do not have to be of the same gender either as this trope can also apply to two people consisting of a boy and a girl.
Energetic and Soft-Spoken Duo is a subtrope. Compare Outside Man, Inside Man, Odd Couple, Cast Calculus (for the evolution of the Red/Blue dynamic through various numbers of ensembles), Tomboy and Girly Girl, Sensitive Guy and Manly Man, Order Versus Chaos, and Elves Versus Dwarves. See also Pink Girl, Blue Boy and Action Duo. If the contrast in personalities between the two actually brings them to blows, you probably have a Tiger Versus Dragon situation. If they are friends it will often be of the Vitriolic Best Buds variety, most commonly where one character is completely oblivious to the vitriol. If race is used to further emphasis these personalities see Salt and Pepper.
Also, despite the name, this trope has nothing to do with that oni.