The Citadel is a large space station in Mass Effect.
This article contains major spoilers for every release of Mass Effect. You have been warned.
The Citadel is located in open space in the Serpent Nebula, with a mass relay nearby. The closest inhabited star system is the Boltzmann system, which is a moderate distance away by mass effect FTL drive.
Little is known about the function of the Citadel in previous galactic cycles, but it is known that it typically becomes the center of the galactic government, and that this function is intended by the Reapers.
In the time period of the games, it is the seat of the Citadel Council, which includes representatives from the major member governments, including the asari, salarians, turians, and humans. The Councilors don't seem to have a great deal of independent authority, needing to consult their homeworld governments for most major decisions, but they appear to be able to appoint SPECTREs at their own discretion.
The true origins of the Citadel are hidden from most of the races in the galaxy. The station was originally built by the precursors to the Reapers at an unknown time in the past, 37 million or more years ago, and may have been further modified by the Reapers since then. Installed in the Citadel is an artificial intelligence (identified in the 22nd century as the Catalyst) designed to keep the peace between organic races and synthetic races. It eventually determined that peacemaking was futile and invented the Reaper cycle as an alternative, turning its own creators into the first Reaper.
Sapient races typically discover mass effect technology left behind by extinct races from previous "cycles", leading them to discover FTL drive and the mass relay network. Discovery of the mass relays inevitably leads races to the Citadel, which typically becomes the center of a galactic government.
The full technological capabilities of the Citadel are unclear, but it is known to be capable of functioning as an enormous mass relay capable of summoning the Reapers from their hibernation area in "Dark Space". It is also the hardware platform for the Catalyst, and it presumably broadcasts signals that control the Reapers themselves.