Prelude
In 2364, the Borg assimilated several outposts on both sides of the Romulan Neutral Zone. Neither the Federation nor the Romulans were familiar with the Borg at the time and were unable to conclusively determine who was responsible for the attacks. It was clear, however, from the level of destruction that the attack was from a species that was by far more powerful than either the Romulan Empire or the Federation. The outposts were not simply destroyed, they appeared as if some great force scooped them up from the planet surface.(TNG: "The Neutral Zone")
It was not until the following year, when the entity known as Q hurled the USS Enterprise-D 7,000 light years to a region of space assimilated by the Borg. An analysis of system J-25 found signs of an advanced civilization, but the cities appeared to have been removed in a manner identical to the loss of the Neutral Zone outposts. A Borg Cube intercepted the Enterprise and drained its shields using a tractor beam. The Enterprise responded by firing phasers at the Borg cube, causing significant damage. The Borg regenerated and repaired its hull while the Enterprise attempted to flee. Unable to outrun the Borg ship, the Enterprise fired again on the Borg Cube, but Borg had adapted to Federation weapons. Several direct photon torpedo hits had no effect on the Borg ship. Just as the Enterprise was about to be assimilated, Q returned the ship to Federation space. (TNG: "Q Who")
Invasion
In 2366, a Borg Cube entered Federation space and assimilated the New Providence Colony in the same manner as System J-25 and the Neutral Zone outposts. Shortly thereafter, The USS Lalo encountered the Borg Cube and reported their position shortly before it went missing. The USS Enterprise-D proceeded to the Lalo's last reported position where they exchanged fire with the Borg Cube. The Borg boarded the Enterprise and assimilated Captain Jean-Luc Picard. Now armed with Picard's tactical knowledge and experience with Starfleet, the Borg ship proceeded to Earth for the assimilation of that planet.
The Federation now considered itself in a state of war. Starfleet mobilized an armada of 40 starships led by Admiral Hanson to intercept the Borg Cube at Wolf 359. The Enterprise was unable to pursue the Borg any further due to battle damage and could not join Starfleet at the Wolf system. Hanson decided that Picard would be considered a casualty of war. The armada would use all its firepower to destroy the Borg ship. No attempt would be made to rescue the captain. The resulting Battle of Wolf 359 turned out to be a disaster. The Federation fleet was decimated. The Borg ship was undamaged. The Enterprise, finally repaired, intercepted the Borg ship and rescued Captain Picard in a daring raid. Using Picard's interface with the Borg Collective, the Enterprise was able to initiate a regeneration protocol that in essence put the Borg crew "to sleep". The Borg Cube self destructed shortly thereafter. (TNG: "The Best of Both Worlds")
Encounters with Hugh
In 2368, the USS Enterprise-D encountered a crashed Borg scout ship with one survivor - a Borg drone nicknamed Hugh by the crew. The Federation still considered themselves at war with the Borg and the Enterprise developed a computer virus that could be implanted in Hugh. The virus would then spread throughout the Borg Collective until the entire collective was destroyed. However, once Hugh was isolated from Collective, he began to shows signs of individuality. Captain Picard no longer felt justified in using a free being as an act of genocide against his own people and did not deploy the virus. Admiral Alynna Nechayev was outraged, stating that Picard's duty as a Starfleet Officer was to safeguard the lives of Federation citizens, despite whatever reservations his conscience may hold. Nechayev also ordered Picard to seize any future opportunity to destroy the Borg. (TNG: "I, Borg")
When the Borg attacked the Federation outpost Ohniaka III in 2369, Captain Picard also questioned his own decision not to deploy the virus. However it was later discovered that the attack was not from the Borg Collective but rather a group of disconnected drones that were following orders from Lore. Once Lore was killed, the attacks stopped. (TNG: "Descent")
Second invasion
The second major battle occurred in 2373. A Borg Cube destroyed the Federation colony on Ivor Prime and was on a direct course for Earth. Admiral Hayes mobilized a Federation fleet in the Typhon sector to engage the Borg. Hayes felt that having a man who was previously captured and assimilated by the Borg on the front lines would introduce an "unstable element in a critical situation". Therefore he ordered Captain Picard to take the USS Enterprise-E on a patrol of the Neutral Zone, in case the Romulans decided to take advantage of the situation. The Federation fleet intercepted the Borg cube and opened fire in the Battle of Sector 001. The Cube quickly broke through the defense perimeter inflicting serious damage and casualties on the fleet. The Flagship sent a distress signal to Starfleet Command, requesting reinforcements. Captain Picard, hearing the distress signal, ordered the Enterprise back to Earth in violation of Hayes' orders. When the Enterprise arrived, the Federation fleet had taken heavy losses. A number of ships had been destroyed including Admiral Hayes' flagship. Many more, including the USS Defiant were severely damaged. Captain Picard took command of the fleet and used his knowledge of the Borg to determine a vulnerable point in the Borg's defenses. He ordered the fleet to target the same section and fire simultaneously. After the fleet concentrated a heavy barrage of torpedoes and phaser fire on the weak point, the Borg Cube was destroyed. (Star Trek: First Contact)