The Conspiracy

Not "conspiracies". Conspiracy. Singular.

—The Question, Justice League Unlimited

The most basic definition of The Conspiracy is "a secretive villainous group with an Evil Plan". Let's break it down:

Secretive. The conspirators always hide their evil agenda or even their very existence. They live as far from a Card-Carrying Villain as possible. As a result, most of them appear benevolent or neutral at first but are then revealed to have been Evil All Along, creating an atmosphere where neither the protagonist, nor the reader knows whom to trust.
Villainous. The conspiracy always plays an antagonistic role to the heroic protagonists. If a conspiracy isn't evil, it's an Ancient Tradition and is discussed elsewhere.
Group. The number of conspirators ranges from a dozens to thousands. Depending on the scale, the conspiracy may be between individuals or entire organizations. Some of them are obviously evil, some appear benevolent, some don't exist at all in the public eye. There may be an Omniscient Council of Vagueness, The Man, The Man Behind the Man, The Dragon, The Mole, and tons of Faceless Goons among them. Conspiracy is like the Lernaean Hydra: every time a head is cut off, it just grows another.
Evil Plan. The conspiracy always has a very specific objective: power, profit, selective destruction, etc. Without this unifying objective, a secretive villainous group turns into The Masquerade.

There are three basic kinds of conspiracies, though the borders between them are fluid:

Ancient Conspiracy. A secret society consisting of otherwise unrelated individuals who pursue some (possibly esoteric) goals. As the name implies, it is pretty old (likely going back to The Knights Templar, The Illuminati, etc.) and very, very well hidden. Their power comes either from long-term strategic planning or mystical sources.
Government Conspiracy. A secret project uniting a small number of members of a powerful but otherwise benevolent organization, usually but not necessarily The Government (hence the name). The project can hardly be older than The Government itself. Their power comes from their Front Organization's resources and the trust that Normal People put into it.
Nebulous Criminal Conspiracy. An informal alliance between seemingly benevolent organizations or individuals and / or obviously evil/criminal/obscure ones. It is likely to be very recent. Their power comes from controlling everything from both sides, though they may be too loosely organized to take full advantage of this. The Syndicate is not a conspiracy in itself because it is obviously evil, but it is almost always a member.

The Conspiracy is generally a great antagonist in any story but two genres make a particularly extensive use of it:

Conspiracy Theory is a Scrapbook Story detailing the progress of Their Plan at different times and places from different perspectives in Faux Documentary style.
Conspiracy Thriller is a more traditional Thriller narrative centering on the conflict between The Protagonist and The Conspiracy.

For further tips on writing both of the above, see Write a Conspiracy Theory. For tropes commonly associated with conspiracies, see The Index Is Watching You.

Multiple concurrent conspiracies form a Conspiracy Kitchen Sink. A character believing in conspiracies is a Conspiracy Theorist. See also Milkman Conspiracy and Conspiracy Religion.

Examples of works/series featuring conspiracies of more than one type:

Anime & Manga

The less known thing about Bee Train's Girls with Guns trilogy is that in addition to playing into three distinct action movie genres (see Bee Train/Trivia), they also each center on a different type of conspiracies:
- Noir pits the protagonists against a Cult-like Ancient Conspiracy of Les Soldats, a secret society formed (probably) by the Cathar survivors after the Albigensian Crusade to oppose the rule of the Roman Catholic Church. Their (or rather, Altena's) Evil Plan, Le Grand Retour, involves reverting the Western society to Medieval norms.
- Madlax revolves around a Nebulous Criminal Conspiracy between The Syndicate of Enfant, the Bookwald Industries, The Government of Gazth-Sonika, and the Galza leadership, who prolong the War for Fun and Profit indefinitely. Also, said war is only a stepping stone in the Big Bad's master plan, which is far more destructive.
- El Cazador deals with the aftermath of a Government Conspiracy called Project Leviathan, a top-secret CIA project to produce Psychic Child Soldiers. More specifically, CIA attempts to cover the whole thing up, which includes killing the few surviving test subjects, one of whom is the amnesiac protagonist.

Eclipse Phase has Firewall, a "heroic" (well, anti-heroic) conspiracy that does not work for any government or criminal organization, and is devoted to the survival of transhumanity at all costs. Though there's also Project Ozma, SETI turned MIB.
- Oh, and Firewall is descended from a fusion of three real world groups JASON, the Lifeboat Foundation, and the Singularity Institute
GURPS Illuminati is an invaluable resource for conspiracy-related games in general.

The Longest Journey series loves to have conspiracies for villains:
- The original game had the Ancient Conspiracy of the Vanguard, a twelve thousand years old Cult led by a DraicKin and aiming to forcibly reunite the technological world of Stark and the magical Arcadia, disregarding the enormous risks to both worlds' existence.
- Dreamfall had Project Alchera, a Government Conspiracy (actually, a corporate one, but corporations ARE the world government in the setting) by the top executives and researchers of the Japanese toy manufacturer WATIcorp to develop and spread Mind Control technology disguised as lucid dream toys.
Deus Ex and it's follow up games (Deus Ex Invisible War and Deus Ex Human Revolution) have these in spades.
Suburban Redevelopment Fund in LA Noire.

The Church Of The Sub Genius blames pretty much everything wrong with the world on "The CONSPIRACY" (which is apparently an acronym for "Cliques Of Normals Secretly Planning Insidious Rituals Aimed At Controlling You").