On April 5, 2063, exactly fifty years from now, Zephram Cochrane will take his warp test aboard the Phoenix, and as a result Vulcans will figure humans are worth visiting.
Cochrane had "dollars signs...money" in mind when he created his warp engine. He doesn't like to fly! He dreams of retiring to a tropical island, filled with naked women. Yet whatever his motivations, Cochrane will prove to be a great man. As he himself will say, "Don't try to be a great man, just be a man, and let history make its own judgment." Yet he will still be surprised when a statue is created in his honor. Geordi La Forge will go to Zephram Cochrane High School!
We know all of this because of Star Trek: First Contact. Cochrane actually makes his first franchise appearance in the original series episode "Metamorphosis," in which we discover how he actually retired. Following First Contact, he became a much bigger element of franchise lore, appearing in "Broken Bow," the first episode of Enterprise, creating the distinctive phrasing of the famous mission statement Kirk and Picard quote in the opening sequences of their shows (while just as famously correcting the dreaded split infinitive by declaring that we'll go boldly). Later, Enterprise reveals that an alternate version of how human's first contact played out created the Mirror Universe when Cochrane fires on the Vulcan and leads a raid on his ship. This occurs in the "In a Mirror, Darkly" two-part adventure.
That may or may not be how it occurs without Picard's intervention. First Contact reveals one last ripple of this famous date in Star Trek history. It begins with humanity's second brush with a Borg invasion. Picard and his crew are initially sidelined thanks to the events of the Next Generation two-part story "The Best of Both Worlds," yet they arrive just in time to save the day, and witness a cube's traveling through time. The Collective believes that pesky humans will be easier to assimilate if they're far more isolated than in the timeline proper. They choose to step in and eliminate Cochrane's flight from history, thereby preventing first contact. It may be worth noting that Cochrane's world is in a state of disarray following the devastation of WWIII (first referenced by Q in "Encounter at Farpoint").
Picard will hardly let that stand. Although things are complicated when Data is kidnapped and almost convinced to join the Borg's efforts, not to mention Riker's exasperated discovery of Cochrane's personality, events play out the way history remembers them (except for the intervention of visitors from the future, much less the presence of the Borg).
Fans who want to follow another thread of First Contact will note another Enterprise episode, "Regeneration," which closes the loop of the movie and may also explain why the Borg were so obsessed with humanity in the first place.
Cochrane isn't alone, it should be noted, in his efforts. He's ably assisted by Lily, who may after all be the idealist history thought Cochrane was, at least before he saw that everything he was told by Riker, La Forge and Troi was true. Yet she has a much more difficult time embracing the strange visitors to her time, possibly because she's brought aboard the Enterprise itself, forced in a real and immediate way to confront everything about the future she helps create. Her scenes with Picard define First Contact even more than Cochrane's.
Fans had a difficult time processing that the pleasant surprise of the Vulcan visit at the end of the movie didn't immediately segue into friendly relations as suggested by the original series. Enterprise features a steady stream of humanity struggling against Vulcan patronizing. Although if you think about it, the signs were all there. We're told by Troi herself that humans were considered too primitive to visit previously, that Cochrane's flight came as a big surprise. Half of First Contact is an emphasis on how decidedly human Cochrane himself is. Spock, meanwhile, is an anomaly in his own crew, not only apparently an exception as a Vulcan serving in Starfleet but also subject to bigotry among his colleagues, the least harmful being his playfully antagonistic relationship with Dr. McCoy. (This pattern of behavior, from oppressor to oppressed, is reflected in the Deep Space Nine continuation of the Mirror Universe saga, by the way.)
All that being said, enjoy this anniversary of first contact!