Amazing Spider-Man is the very first comic series although Spidey himself first appeared in Amazing Fantasy #15.
In 1972, Marvel had put out a title called Marvel Team-Up which always featured Spidey teaming up with various Marvel characters, kinda like how Avenging Spider-Man is today. It was looked at as the official 2nd Spider-Man ongoing.
In 1976, they launched Peter Parker The Spectacular Spider-Man which eventually in 1988 became just The Spectacular Spider-Man.
In 1985, Marvel cancelled Marvel Team Up and launched Web of Spider-Man. All 3 titles had their own plots, stories and themes but were joined strongly continuity wise and occasionally crossed over (like in Kraven's Last Hunt).
Eventually Todd McFarlane came along in the late 80's as the artist on ASM and became a superstar. In return, Marvel launched the adjective-less "Spider-Man" in 1990 as Spider-Man's 4th ongoing. In 1996 it was renamed Peter Parker: Spider-Man to celebrate the end of the Clone Saga and Peter's return to the costume.
All 4 books lasted well into the Clone Saga when Ben Reilly became Spider-Man. Web of Spider-Man was cancelled and replaced in 1995 by The Sensational Spider-Man as a vehicle for Dan Jurgens who was writer and artist. These 4 titles lasted all the way to 1999 when the entire line was revamped and condensed to 2 titles, ASM and Peter Parker: Spider-Man.