Nicholas Hammond (who was one of the VonTrapp children in "The Sound of Music") played the protagonist, Peter Parker/Spider-Man. The TV show producers/writers took several liberties with adapting the series, compared to the original Stan Lee/Steve Ditko comic book stories. In the original comics, Peter Parker was a high-school student when he got his powers. Here, Peter Parker is already a graduate college student. He's still a budding scientist, and a chem-lab experiment using radioactive waste is where the fateful spider comes along to bite the hero. He gains wall-crawling ability, super-strength and a special "spider-sense" which is basically ESP that focuses on nearby danger. Shortly afterwards, he uses his science skills to create a webcasting device that he wears on his left wrist (comics purists note the glaring fact that it is worn on the outside of his costume, and not underneath.)
In the pilot episode/movie, Peter berates a purse-snatcher while perched vertically on a nearby wall-- the shocked thief is stalled just enough for the police to arrive. Local newspaper publisher J. Jonah Jameson catches wind of the alleged "Spider-Man" and lets it be known that anyone who can get photographs of him will get paid handsomely. So Peter designs his spider-costume, and goes about photographing himself all around town. Jameson is none the wiser, and though he thinks Parker is a little flaky, he can't deny the authenticity of the photos.
Shortly, a bizarre crime-wave hits NYC, which will prove to be Spider-Man's big challenge. It seems that a corrupt self-help guru with a large clientele has an elaborate mind-control device which he is using to hypnotize his clients. Broadcasting submliminal messages with membership pins that double as receivers, his clients, from various ordinary walks of life, start committing bank robberies; during the getaway, they always crash the car at a predetermined location; a group of henchmen retrieve the money, and the unwitting robbers are left to die or face prosecution, with no memory of what happened.
The police investigator following the case is the hard-nosed Captain Barbara. He's skeptical of why Peter tends to show up at scenes of trouble, usually in the aftermath of Spider-Man having been there. Spider-Man's investigation leads him to the guru's headquarters where he must fend off a group of martial arts guards armed with bo staffs. Peter's science skills lead him to find out what's causing these otherwise normal people to turn crooked, and the climax is set.
Further episodes would continue to explore the adventures of Peter Parker and his costumed alter ego. David White (Larry on "Bewitched") plays Jameson in the pilot. Robert Simon played Jameson after the pilot. Other supporting characters included Daily Bugle secretary Rita (possibly loosely based on the Glory Grant character in the comics), Aunt May (who only made sporadic apparances), and Julie, a female "rival" reporter (who was not in the comics, but seemingly created as a "Lois Lane" and potential love interest).
The scripts kept the characterization of the hero intact (morally upright, though he frequently has bad luck, and half the people in Pete's life think he's kind of a flake), and Hammond as Peter/Spidey was very good, though he was clearly playing an adult version of the character-- it would have been interesting to see an adaptation of the original high-school setting of the strip.
The TV show never featured any villains from the comics series, instead focusing on human crime lords and rogue scientist-types (curious, then that the Wilson Fisk/Kingpin character was never attempted).
Despite taking place in New York City, 99% of the series was actually filmed in Los Angeles (except for some establishing shots and stock footage). The special effects, though perhaps adequate for the time (and likely, a small budget), come off as very dated in today's CGI-saturated culture.
Stan Lee had a "consultant" position on the show, which, unofficially, was likely a token role, as in later interviews Lee would claim to have no practical influence on the creative direction or the execution of episodes.
The series apparently changed time-slots several times during its run, which probably didn't help ratings. Compared to the "Incredible Hulk" TV series, "Spider-Man" was less well-executed, though in fairness they did have a larger cast to deal with and a fixed thematic venue (New York) to try and contend with.