What is a Conspiracy Card and how does it work?
Conspiracy is a special, standalone set in MtG, currently consisting of two expansions: Conspiracy; and Conspiracy: Take the Crown.
First and foremost, I'd recommend you read these articles, which explain some of the set and its mechanics:
The Conspiracy format
The Mechanics of Conspiracy
The second article explains both what are Conspiracy cards and how Hidden Agenda works. Also, this is the official rule for Conspiracy cards:

905.4. At the start of the game, before decks are shuffled, each player may put any number of conspiracy cards from his or her sideboard into the command zone.

And here are the rules for cards with Hidden Agenda:

702.105a Hidden agenda is a static ability that functions as a conspiracy card with hidden agenda is put into the command zone. “Hidden agenda” means “As you put this conspiracy card into the command zone, turn it face down and secretly choose a card name.”
702.105b To secretly choose a card name, note that name on a piece of paper kept with the face-down conspiracy card.
702.105c Any time you have priority, you may turn a face-down conspiracy card you control in the command zone face up. This is a special action. Doing so will reveal the chosen name. See rule 115.2h.
702.105d Hidden agenda and another ability of the object with hidden agenda that refers to “the chosen name” are linked. The second ability refers only to the card name chosen as a result of that object’s hidden agenda ability. See rule 607.2d.

(Mostly) Unrelated to Conspiracy is The Commander format, from which the term Command Zone is taken. The Command Zone is a special zone in a game of Commander, where the commander (or general) of your deck begins the game.
Some Conspiracy cards reference the Command Zone, as it's a zone in the game that's reserved for objects with overarching effects on the game. See rules 408.1...

408.1. The command zone is a game area reserved for certain specialized objects that have an overarching effect on the game, yet are not permanents and cannot be destroyed.

And 408.3:

408.3. In the Planechase, Vanguard, Commander, Archenemy, and Conspiracy Draft casual variants, nontraditional Magic cards and/or specially designated cards start the game in the command zone. Each variant has its own rules regarding such cards. See section 9, “Casual Variants.”