The exact origins of the Borg are unknown except that they originate from the Delta Quadrant of the galaxy. The first time any craft comes into contact with the Borg is when the Omnipotent Q teleports the crew of the Enterprise to encounter a Borg Cube. Borg spacecrafts are giant shapeless cubes. The Borg Cubes are highly advanced after having assimilated many cultures thereby gaining the assimilateds knowledge. The Borg have built transwarp machines which have the ability to transport cube to other parts of the galaxy and not just to other solar systems. It is one of these transwarp cubes that Voyager ultimately uses to return to Earth.
The Borg are similar to the Cybermen in the Doctor Who entertainment Universe. The Borg don't reproduce like humans do, they assimilate, i.e convert them into a Borg. Once you become a Borg, you loose all individuality and sex drive, you work for the Borg Collective. The Borg will assimilate young and old as they did with the young Seven of Nine. Unlike Cybemen, the Borg don't replace the whole body, just parts of the body. As you can see from Seven of Nine, Voyager's Doctor was able to return a Borg to a near human state with just a few Borg components still left on.
The Borg when hit by phaser fire, adapt and the other Borg drones in the group become immune to that phasers frequency so the crew have to change the frequency to be able to disarm another one. In some of the Borg Cubes, there is a Borg Queen who controls the hive. The Borg Queen appeared in the First Contact and also in episodes of Star Trek - Voyager. The Borg Queen was played by the same actress both times, Alice Krige.
Any undefended starship hearing 'We are Borg, prepare to be assimilated', knows that they are in trouble. Sometimes the Borg will ignore a ship such as in the case of Seven of Nine's previous life as a human. Seven's parents shadowed a Borg Cube for a long period but assimilation was not done until the Borg decided that they were a threat.
The Borg being not satisfied with our universe succeeded in opening a portal to another parallel universe, a fluid universe. In the other universe is Species 8472, an insect like creature. Voyager and a Borg Cube agree to work together to stop the Species 8472 from destroying Voyager and the Borg.
In First Contact, the Borg travel back in time to when Earth was about to make first contact with an Alien race, the Vulcans to prevent it from happening. The Enterprise is caught in the same time slip and are able to assist Zephram Cochrane and his engineers in defeating the Borg and making First Contact.