Classic X-Men (1975-83)
Noteworthy Comic Panels
Chris Claremont took over the writing duties for Uncanny X-Men shortly after the reboot (Giant-Size #1). Claremont altered the nature of comic book narratives and this could be seen in obvious ways (recurring epic villains in almost every issue) as well as in subtle ways (flashbacks/recaps were integrated into the narrative with nuance and they didn't always appear at the start of issues). The new team was grittier and more mysterious, while Cyclops and Jean Grey stayed on to provide guidance and experience. During these years, Claremont did everything right with the X-Men and I have gathered some notable panels that illustrate some of the best features from the classic X-Men era.
Gellin' like Magellan
There were a lot of awkward relationships with the original X-Men, mainly because Warren and Scott were fighting over Jean. The new team was very independent personalities from around the world however they came together quickly and bonded as a team. Each issue would explore those relationships and sometimes action spoke louder than words. We knew that Colossus and Wolverine were getting on well when they started combining their strengths for unique combo moves. Additionally, Storm was always more of an older sister figure and Claremont avoided hokey romantic triangles for the African goddess. The soap opera elements didn't infect X-Men titles until the early 1990s.
The New Old Guard
The original X-Men were considered teenagers full of over-enthusiasm and angst. Those relationships could be silly and it was often embarrassing to root for a superhero team that constantly screwed up and had to be bailed out by their professor. With the new team, each member had already developed professional competences and some of the members arrived at the mansion as cagey veterans when it came to leadership roles or shouldering serious responsibility. The new team's demeanor and maturity made more sense for mutants who were considered an oppressed and feared underclass. It wasn't easy performing heroic deeds for those who underappreciated you, and so it made sense that only wise, exotic personalities would be able to handle it. Additionally, interesting dynamics were forged between the original X-Men and the new team because Cyclops wasn't leading a bunch of kids and his orders were not blindly followed.
Divide and Conquer
Claremont was a storytelling genius and would jump right into the action. One notable example of this is when suddenly the X-Men are performers in a circus freak show. Beast arrives at Xavier's Mansion and there is no sign of the X-Men which leads him on an investigation. It is ironic that Beast is the one to discover the X-Men at the circus because he is one of the mutants that is most often treated as a freak by regular humans. Beast uncovers the nefarious machinations of the mastermind, Mesmero. It was Mesmero who had captured the X-Men and set them up in the circus however there is always a bigger fish. Magneto has been manipulating the situation and Mesmero is a mere pawn in the master of magnetism's game. The action for these issues is exciting and there are a lot of twists and turns however the most brilliant narrative development is when Beast and Jean are separated from the others during their escape from Magneto's underground facility. Beast and Jean (Phoenix) come to believe that the new X-Men have perished, meanwhile the new X-Men land themselves in the Savageland and also believe that Beast and Jean didn't make it to the surface safely. This sets up the necessary conditions for Jean to transform from Phoenix to Dark Phoenix. The story line wasn't perfect because there are many moments where the reader can't understand why upon returning to Xavier's mansion the new X-Men haven't contacted the relevant parties (Avengers, Moira MacTaggert) to find out about what happened to Jean and Beast. That is a major contrivance that is picked up by the critical reader.
If You Want My Body, and You Think I'm Sexy...
The Western world was experiencing radical social changes by the mid-70s where the taboos on sex and sexuality were loosening fast and furiously. Disco was taking over in the clubs and it was all about flashy, tight-fitting outfits. On the other side of things, rock n' roll was becoming more glamorous and fostering groupie culture. Everyone was banging or at least pretending to and certainly aspiring to. Eventually, HIV popped-up and quickly caused society to question whether being a self-righteous whore was the best future for the species. The sexual revolution was dampened but from 1975-83 there was no holding back. This liberal attitude toward sex pervaded comic books and artists like John Byrne didn't hesitate to exploit the era whether through depicting scantily clad tribesmen in the Savageland or when detailing the risque fetish outfits of the Hellfire Club.
The original X-Men team were pretty chauvinistic and Jean Grey was rendered to chattel for the satisfaction of their machismo. With the new team none of that was a factor because Jean had become incredibly mature and competent through her fusing with the Phoenix entity, while Storm had already been a leader and had lived as a goddess in her native Africa. Ororo (Storm) had been orphaned as a child in Egypt and quickly learned how to fend for herself. Storm could do anything that the men could do, and often better. She was a competent fighter and often bested other mutants whose powers were actually related to hand-to-hand combat (Callisto and Crimson Commando). It made a lot of sense to develop Storm in such a way that she could take over as the leader of the X-Men. She had the right stuff.
Not Going Away Anytime Soon
In the first issues of X-Men it was established that mutants were distrusted and often loathed by regular human beings. The original X-Men often fought against non-mutants and for a time, to be a mutant seemed like a rare condition. However, the popularity of the new X-Men team resulted in a rapid expanding of the mutant population within the Marvel universe. Sometimes, it seemed like every other character was a mutant. When the Morlocks were discovered there was a true sense of how common the mutant X-gene would become. Eventually, mutants were culled or controlled (Mutant Massacre and Genosha) so that it was once more an elite phenomenon to have mastery over your mutant powers. The advantage to spreading the mutant condition was that new superhero teams could be assembled. Eventually, Moira MacTaggert had an entire team on Muir Island and they could jump into action when needed (later they became the new government-sponsored X-Factor). If Avengers were too pristine for your tastes then by the mid-80s you had new options for sticking strictly to mutant-based comic book titles (X-Factor, Alpha Flight, New Mutants, Wolverine, and Excalibur).
Off with Their Heads
The X-Men were finally on the map for Marvel during the Dark Phoenix Saga. It was the first time that events for mutants were affecting the entire universe. On many occasions, Magneto had caught the attention of the Avengers or Fantastic Four but Magneto's evil never impacted cosmic beings or the universe as a whole. The transformation of Jean from Phoenix to Dark Phoenix was an immaculate conception which played out for over two years.
Friends in High Places
The new X-Men team was composed of exotic mutants from around the world. Claremont explored the backstories for the new X-Men which provided ample opportunity to create new characters, especially mutants. Wolverine came from the Weapon X program in Canada and as a result it made sense to create a mutant superhero team based in Canada (Alpha Flight). Those characters were legitimated through their connection to Wolverine, but Wolverine's past could be explored through relationships with those new characters. Storm already knew Shadowking from her days as a child thief in Cairo, Nightcrawler was blood related to Mystique, Colossus's little sister Illyana became a founding member of the New Mutants (Magik), and Banshee would eventually join Moira and her band of reluctant mutant heroes on Muir Island. The backstories for the new X-Men were creating an intricate web of activity for mutants in the Marvel universe.
If you Gotta Go... You Gotta Go
Not all of the X-Men fit in right with the team. Thunderbird was killed off almost immediately while Banshee was injured and retired. Claremont was very attentive to the team dynamic and wanted only essential ingredients for the X-Men. Once Rogue had proven successful as a neophyte X-Men, it was becoming clearer that Kitty Pryde didn't fit in with the team properly. Although, she could have been a better leader for the New Mutants than Dani Moonstar, Marvel decided to not demote Kitty. The idea was that the New Mutants would become an even more brooding and grimy group than the X-Men.
Meanwhile, Kitty was well-established as a sweetie. Even the Kitty Pryde & Wolverine mini-series didn't have anyone convinced that Kitty had what it took to be physically or mentally tough. The best that Marvel could muster was to have Kitty become emotionally tough, but only through expressing emotion (consistent with her established character). This meant that she required a foil - a character always on the verge of breaking down emotionally. Rachel Summers (Phoenix) served that purpose and eventually Kitty and Rachel would leave the X-Men to form Excalibur. Excalibur was the tofu version of the X-Men and the drama was based around soap opera-like emotional turmoil sooner than being about gritty resolve in the face of physical brutality and political adversity. Moving Kitty out was a good move although I believe that Rachel's character could have been modified in such a way that she would have fit with the X-Men. The problem was that Rachel and Rogue had similar backstories and demeanor. Rogue was the one who toughened up while Rachel remained flighty and fragile.
Professor Charles Xavier was the founder of the X-Men, but he had always run to the rescue and bailed out the old team. The new team had to be more independent which rendered Prof. X to a third-wheel role (no pun intended). During the Dark Phoenix Saga, the X-Men are introduced to Lilandra and the Shi'ar Empire. Prof. X falls madly in love with Lilandra and decides to stay with her. When that motivation didn't sell, Charles was psychologically injured stayed with Lilandra and her medical bots in a galaxy far far away.
Shipping out Charles was the right move because it granted greater independence to the X-Men, but especially to the New Mutants. Additionally, Magneto's character became vastly more complex once Charles pleaded that Magneto take over the headmaster duties at Xavier's school. Later, there would be convoluted explanations for how Magneto had been willing to turn away from evil and become a hero (ironically this too involved the Sh'iar Empire through the machinations of Erik the Red). Xavier venturing off with Lilandra and the Starjammers allowed the X-Men to become fugitives. It never would have made sense to have Xavier disgraced on Earth and if he had failed to redeem his X-Men then he would have been viewed as impotent and useless. With the new team of X-Men, Xavier had to bow out of leadership and mentoring duties.
Bonus! - Cameos
The cameos of the old X-Men in issues of Avengers, Fantastic Four, or Amazing Spiderman were generally lackluster and usually involved the X-Men being portrayed as young fools in over their head in the superhero game. In the classic X-Men era with the new team, cameos were something special and the X-Men were often there to bail out other heroes under duress. In cases of mistaken hero identity, the X-Men didn't come off as fools but instead seemed totally bad ass and overwhelming - the heroes were lucky that the X-Men realized their mistake and backed down. These are some notable panels from non-X-Men comic books that featured X-Men characters (heroes and villains).