The Borg represent one of the most serious known threats to the Federation, and indeed to all species in this galaxy.  Little is known about their origins; their combination of biology and technology has been evolving for thousands of centuries.  At some point they began to 'network' their minds together into a single species-wide consciousness - the Borg Collective.  Unfortunately, this collective mind had little regard for individuality; it embarked on a program to assimilate into itself any species it considered useful, by force if necessary. 
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There are many variations between Borg drones, optimising them for their various roles,  but most drones share certain common features. Their bodies are covered with a leather-like substence crisscrossed by varius tubes and mechanical appliances. One eye - most often the left - is usually replaced with a holographic viewing system,  while the right forearm is often replaced with a mechanical appendage.  Borg have a built-in replicator unit which can create or modify their mechanical systems as required and synthesize nutrients for their organic systems. Drones carry a supply of nanoprobes which they inject into other lifeforms; these nanoprobes rapidly begin to construct machinery in the victim, beginning the assimilation process.  The Borg rely on an onboard power cell for energy, and this must be recharged periodically.  The two features shared by every known Borg are a shield system and a subspace transceiver which keeps the links between the drones intact; it is this network of links which comprises the collective itself. 
In 2364 a Borg vessel is known to have passed through both Federation and Romulan space, destroying outposts of both powers. The Borg chose not to attack the Federation at this time,  presumably because they where on their way to some other more urgent location. In 2365 the Enterprise-D was flung into system J-25 by Q, where they encountered a Borg vessel. The Borg sent scouts to probe the Enterprise - the first of these was quickly killed, but subsequent drones proved to have a shield device which was immune to phaser fire. The Borg ship subsequently attempted to assimilate the Enterprise, but their cube was quickly disabled by the Federation vessel. However, an away team sent to the Borg ship discovered that the collective was rapidly regenerating its ship. The Enterprise attempted to depart the area before further conflict ensued, but the borg gave chase. As with the drones, further attacks by the Enterprise proved to have no effect on the Borg vessel.  Later analysis showed that when the a Borg is attacked the collective mind analyses the destruction of individual drones and creates a defence, which all drones then add to their own systems  via their replicators. Their ships operate on similar principles.  The Borg vessel nearly destroyed the Enterprise, but Q intervened to save the ship.  However, the Borg have displayed a good degree of persistence and since this time have launched two invasions of the Federation - one in 2366  and once in 2373 - causing massive loss of life before being narrowly defeated. 
Recently the Borg have encountered the life form they refer to as "Species 8472". Species 8472 proved both to be invulnerable to Borg nanoprobes and to have superior technology. In response to a Borg invasion of their space species 8472 nearly destroyed the Borg;  with the help of modified nanoprobes from the USS Voyager the Borg where able to force Species 8472 to disengage and retreat from the conflict. 
Voyager had several subsequent encounters with the Borg.  On one occasion the Voyager crew was able to spread a computer virus amongst the Borg which enabled drones to rebel against the collective, causing great disruption to the group mind,  In 2377 Janeway used a Borg transwarp hub to finally return Voyager to the Alpha Quadrant. The hub and a large section of a Borg Unimatrix was destroyed in the process.