With a little help from Wikipedia... Here's a capsule summary of Borg history.
ST: TNG- Q Who?: Borg first appear when Q flings Enterprise D across the galaxy so Picard can learn humility (or something like that). It's suggested that the Borg may have been responsible for the destruction of Federation and Romulan colonies in the season 1 finale "The Neutral Zone".
- The Best of Both Worlds (two-part season 4 finale): Picard is abducted by the Borg and turned into Locutus of Borg to serve as the Borg's representative to humanity ("resistance is futile..."). The Borg use Picard's knowledge of Starfleet and decimate 40 starships, including Klingon ships, at Wolf 359. The Enterprise D crew captures Locutus, learns information from him that allows them to destroy the Borg cube, then removes the Borg implants from Picard.
- "I, Borg" (fifth season): The Enterprise D crew rescues a lone Borg that Geordi names "Hugh". There's a big moral dilemma because they had planned to use Hugh to infect the Borg collective with a computer virus, but after being separated from the collective, Hugh shows signs of independence. They decide to let him return to the collective virus-free with his individuality intact. They hope his individuality will infect the Borg instead.
- Descent (sixth season): A group of rogue Borg who have "assimilated" Hugh's individuality. Sadly, they have fallen prey to Data's evil older brother Lore. Lore manipulates them hoping to turn them on the Federation whilst also corrupting Data's ethical subroutines through an emotion chip he stole from Noonien Soong, Data and Lore's creator. Data's ethical subroutines are restored, Hugh leads the renegade Borg into attacking Lore's complex, Data fights with and deactivates Lore. Data takes the emotion chip because it's the only way to safely dismantle Lore. Hugh assumes leadership of the renegade Borg and Hugh's storyline is dropped, never to be seen again.
- ST: First Contact (ST VIII movie): After failing to assimilate Earth in battle, the Borg travel back in time to stop Zefram Cochrane's first warp speed flight which resulted in first contact with the Vulcans. The Enterprise E (the Enterprise D was destroyed by Riker in the previous film ST: Generations) follows the Borg back in time. Our heroes win the day, the time line is restored, and the Enterprise E gets to peak in on history as Cochrane meets the Vulcans. Everybody dances. However...
ST: EnterpriseRegeneration: What the Enterprise E crew fails to realize while back in time, along with the viewers because this was totally retconned, is that part of the Borg sphere crash lands in the Artic, whereupon it's discovered in Archer's time. Once it's unearthed, the Borg revive and steal an unarmed research ship, modifying it to match Starfleet technology. Before the crew of the NX-O1 destroys the Borg, they manage to send a transmission that will reach it's destination in 200 years. Thus by the time of "Q Who", the Borg are already on their way to Earth. The Borg never identify themselves as Borg, but various characters recall that Zefram Cochrane used to claim that cybernetic creatures from the future tried to prevent first contact with the Vulcans. Of course, since Cochrane was frequently drunk, no one believed him.I'm not even going to touch the mess that is ST: Voyager. However, it's in Voyager we learn that the Borg have the ability to open up worm holes so they can travel very, very fast. Also, somewhere we learn that it was the Borg that wiped out Guinan's people. Guinan, and the rest of the survivors of her planet, were of course rescued from a glowy space-time ribbon by Kirk in ST: Generations. Kirk fell into the ribbon and was in turn rescued by Picard only to die at the hands of Alex DeLarge, I mean, Malcolm McDowell.
Later on, in a novel co-authored by Shatner, Kirk's body is stolen by the Romulans who bring him back to life with a Borg regeneration chamber... but that's neither here or canon.
Somewhere in the mess that is ST: Voyager, we learn of an expedition that was sent out to find and investigate the Borg sometime before the events of Q Who?, possibly on the basis of Cochrane's rumors (more retconning). Oh, and I'm not sure if you noticed, but there's incredibly hot Borg on the crew of Voyager (not to mention the holographic Doctor who travels back in time to impersonate an annoying high level bureaucrat, constantly interfering with McGyver's attempts to beat the Federation to the warp drive by a couple hundred years. Ironically, McGyver does a much better than job than Kirk and Picard combined at protecting the universe from people with the forehead-deformity gene.