Amazing Spider-Man worked as a title because the main character's life was interesting to readers whether he was swinging through the city as Spider-Man, or not: the more mundande soap opera of Peter Parker's loves, friendships, and conflicts were often, amazingly enough, at least as interesting as the super-heroic happenings. During Ditko's run as artist (#1-38) saw Peter as a high school kid, skinny and unpopular, beset by the same problems that his readers faced - i.e., getting picked on by bullies, trying to speak to the prettiest girl in class, worrying over tests, etc. Of course, he was different from ordinary kids in that he had super-powers and led a secret life - so the bullies really didn't bother him so much, at least. But unlike Batman or Superman, or any of the other traditional heroes, Spider-Man/Parker often found his life complicated by his extra-normal existence. As a result, Peter Parker often found himself frustrated and ridden with self-pity. Self-pity - in a superhero?! Such a thing was unheard-of in the early 1960's, but in ASM it took place issue after issue - and readers loved it.
In the course of normal super-hero business, however, ASM wasn't so offbeat a title that lovers of the genre couldn't find anything to like about it. Spider-Man quickly found himself fighting against a Rogue's Gallery of the weirdest and most interesting villains to appear in the comics, rivalling even Batman's or Dick Tracy's enemies. Some of them were, like himself, victims of weird accidents involving energy or radioactivity (Doctor Octopus, Elektro) of some sort; many were just petty crooks with gimmicks (the Vulture, the Green Goblin). Some of them were monsters one moment and friends the next - like Dr. Curtis Connors, who was Spider-Man's ally and scientifically-able assistant when normal, but who could turn into the deadly Lizard under the wrong conditions. Like Spider-Man himself, some of his villains had secret identities - and the revealing of those identities would prove to be major events.
When Ditko left the series, his chores were taken over by John Romita Sr., who brought a smooth, polished artistic style, honed over the years at both Marvel and DC, doing everything from Daredevil to romance comics. Romita's fight scenes were like action movie stills; his characters radiated emotion and energy. His men were dangerous and his women were breathtakingly beautiful. Romita had been on the title only a few months, in fact, when the love of Peter Parker's life, Ms. Mary Jane Watson, was introduced - she had lurked in the background of a few issues before being fully revealed in #42. With her, she brought the title into the mid-to-late 60's with her fashion sense, her 60's lingo, and her modern sensibilities in general. The series got its first black supporting character (Joe Robertson, Peter's colleague at the Daily Bugle) and the harmful effects of drug use were shown - both very topical subjects. As a result of all this, ASM became a title that moved with the times, not in spite of them.
Arguably, this would become the most creatively fertile period in the title's history. With all of its characters in place, and its conflicts and situations set up, Amazing Spider-Man could be counted on month after month for razor-sharp storytelling. Stan Lee could, to his great credit, pack a lot of plot and action into a single issue's pages, and it was during this time that he was at the height of his powers on this series. Some issues were almost entirely given over to battles; some contained pages that seemed to sag beneath the load of words they must bear. But whether wordy or not, Lee was pouring everything he knew about comic book scriptwriting into the book. As a result the fictional milieu that the characters inhabited became almost as familiar to readers as the one they inhabited in the 'real' world.
Beginning with issue #111, Gerry Conway took over as writer as Stan Lee went on to pursue other things. Romita would continue as the sole artist for some time - his workload sometimes eased with the help of Gil Kane - until with #125 Ross Andru (who had worked on The Flash in the 1960's) took over as penciller. Conway and Andru would make a good team - if not on par with Lee/Romita, they at least would turn out stories as good as anything else Marvel was printing in the 70's. It was during their tenure that both the Green Goblin and Gwen Stacy were killed off, and such characters as the Tarantula, the Jackal, and the Punisher were introduced. The decade would finish with the title being created under different hands as writers such as Len Wein and Marv Wolfman, and artists like Al Milgrom would continue the Wall-Crawler's adventures.
The title would continue until 1998; but, we'll leave that tale for another time. Pop-Cult.Com covers any and all decades, but right now we're leaving the emphasis on the 1960's and 70's - when Amazing Spider-Man was still on top.
You can see a gallery of Amazing Spider-Man covers from #1 to #200 here.