The universe of Star Trek is filled with a vast array of friends and foes to the various captains that act as the protagonists throughout the franchise. These tend to vary from series to series, switching between individual characters to entire races.
The Original Series had the Klingons, but Kirk had the infamous Khan. Deep Space 9 had the Dominion, but the morally ambiguous Captain Benjamin Sisko's specific nemesis was Dul Dukat, a pesky and slimy Cardassian who was constant pain in the backside for the poor captain. But what about Captain Picard from The Next Generation? The answer changes depending on who is asked, and it makes for an interesting study into what the TV shows were trying to suggest, and what the movies wanted.
When the aptly named movie Nemesis came out, Paramount were desperately trying to recreate the good old days of Star Trek, and felt the need to introduce an antagonist with the same Khan level of rivalry for captain Picard. They introduced Shinzon, a Romulan-made clone of Picard himself, played by a very young Tom Hardy. The Romulans created him using some ill-gotten DNA from the dear captain, in a convoluted plan to replace Picard with their clone agent. Things go awry, as they often do, with Shinzon becoming the leader, but what Paramount were trying to do with his character was create the ultimate nemesis for Picard.
For a lot of fans, however, they failed to carry this through. Shinzon felt like just another one-off villain, having no pre-established history with the other films or TV shows to hammer in his deep connection with Picard. Khan at least was introduced in The Original Series prior to the film, so there was an existing understanding of his backstory and how he personally connected with Kirk, before the film even started.
The nemesis feeling that Paramount were trying to create (or recreate) was overshadowed massively by the preceding of various other, pre-existing nemesis characters in Picard's life, and at the top of this list was none other than the Borg. The Borg present an interesting dilemma of the nemesis crown. While normally this title would be given to an individual, like Khan, Gul Dukat, or even Shinzon, the Borg, by their very nature, lack any sense of individuality. Instead of his nemesis being an individual of the Borg race, it was the Borg as a collective, a confusing notion that most likely led to the attempt to bring in Shinzon. However, this also led the show's creators to introduce the Borg queen, something that still does not particularly sit well with a lot of fans who believe she ruined what the Borg were supposed to be.
The Borg collective could be seen through The Next Generation as the nemesis of the Federation, not just Picard. Howerver there is an element of personal conflict when it comes to being a nemesis. The Federation fought against the Borg out of a necessity to survive and protect themselves, but it was never personal like it was with Picard, who was assimilated and transformed into Locutis of Borg. While he was eventually rescued, his assimilation was traumatizing, and resonates throughout not only the rest of the show, but the movies and the modern Picard show too. It even had ripple effects over into Deep Space 9, being a pivotal background story for Sisko, whose wife was killed by Borg under Locutis’ command.
While Shinzon and other villains from the franchise riled Picard up, tortured or hurt him, made him angry or provoked various other responses from him, the Borg were the only ones to truly get into the captain’s head (literally) and result in one of the few time audiences see the normally stoic captain break down. The episode “Family” is regarded as one of the best episodes of TNG for many reasons (despite its negative critique from Roddenberry), and right at the top of these is the emotional moment Picard breaks down into tears. The quote below sums up his trauma best:
“You don't know, Robert. You don't know... They took everything I was. They used me to kill and to destroy and I couldn't stop them. I should have been able to stop them. I tried...I tried so hard. But I wasn't strong enough! I wasn't good enough! I should have been able to stop them, I should've, I should...!”
This is not the only moment of Borg-related emotional outburst Picard shows, the other being in the First Contact movie where he loses his temper and smashes a display case. He literally screams ‘NOOOOO!’ and then makes a passionate speech about how much he hates the Borg. It’s even mentioned after this point by the character Lily that the Borg are Picard's ‘White Whale,’ in reference to Moby-Dick’s obsession driven hatred for the whale.
"They invade our space, and we fall back. They assimilate entire worlds, and we fall back. Not again! The line must be drawn here! This far, no further! And I will make them pay for what they've done!"
These two moments are the only two examples of any villain in Picard's story to really trigger such an emotional reaction from him. He shows irritation with Q, and a fair amount of anger too (though nothing in conspired to the contempt Sisko showed for him), but never has the same triggered response.
The writers stuck gold in these moments, masterfully knowing when to leave it alone, and when to really show the traumatic effect the Borg had on the captain. They have written this well, always suggesting it’s bubbling under the surface, seeping through the stoic facade the captain so desperately tries to maintain, until it’s all too much, and he breaks down in a very realistic way. The Borg are his true nemesis, and one he never fully manages to defeat until he is able to forgive and move on from what they did to him, something which is explored more in Picard season 2.