Stealth ("Big Time") Suit
Technically, this suit is known as the Stealth Suit. The "Big Time" moniker comes from the overall publishing initiative that the Amazing Spider-Man comic series was undergoing at the time. Like the Spider Armor Mk 2, this was a suit developed specifically to deal with one particular villain, so you'll only see it in action if you read the Kill To Be You arc, comprising issues 648 - 651.
Spider Armor Mk 3 Suit
First appearance: Amazing Spider-Man #682 (2012)
Developed to handle the supergroup of villains known as the Sinister Six, the Spider Armor Mk 3 has a ton of abilities that make Spider-Man less of "a guy with the powers of a spider" and more "basically Iron Man + Batman." To see all of its ridiculous arsenal in action, read the six-part storyline Ends Of The Earth.
Spider-Man 2099 Black Suit
First appearance: Amazing Spider-Man #365 (1992)
Another suit not worn by Peter Parker, Spider-Man 2099 is actually Miguel O'Hara, a geneticist living in New York in the year 2099. Miguel has traveled back and forth from his time to ours on several occasions, but his full-fledged debut is still the best. Check out the original Spider-Man 2099 run from 1998 and enjoy.
Iron Spider Suit
First appearance: N/A
This one's a bit of sticky wicket - the design itself is copied over from Avengers: Infinity War, but that was, itself, inspired by the Iron Spider of the comics. The comic version debuted in Amazing Spider-Man #529, in the middle of the Civil War storyline. That crossover series hasn't aged particularly well, but Spidey's adventures are a bright spot among some really dour storylines. Check out The Night the War Came Home for a particularly powerful series of events and some Iron Spider action.
Velocity Suit
First appearance: N/A
This suit was specially designed just for the Spider-Man PS4 game, though it does have some comic influences - it was envisioned by artist Adi Granov. You can see his work on several Marvel projects, but one particular favorite is the one-shot Dark Reign: The Cabal.
Spider Armor Mk 4 Suit
First appearance: Amazing Spider-Man Vol. 4 #1 (2015)
After Peter Parker got his body back from Doctor Octopus (it's a long story), he used his newfound riches to develop the Spider Armor Mk 4. Unlike previous Spider Armors, this one actually became a regular suit for Parker to wear, which he did for a few years. You'll find this outfit in most of the Amazing Spider-Man Vol. 4, but pick up Amazing Spider-Man Vol. 4 #1 to see just how ridiculously awesome it can be when Parker decides to go full James Bond.
Spirit Spider
First appearance: Amazing Spider-Man Annual #38 (2011)
For some reason, this costume is called "Spirit Spider," but is clearly based off "Ghost Spider" from the comics. Ghost Spider doesn't show up much outside of his debut and I don't want to spoil how he comes to be, so go pick up Amazing Spider-Man Annual #38 and see for yourself.
Spider-Man 2099 White Suit
First appearance: Spider-Man 2099 Vol. 3 #1 (2015)
Hey, just because he's a spin-off character doesn't mean he can't have a costume change of his own. Yes, this is still Miguel O'Hara, with a redesigned suit that trades the dark and menacing color scheme of his original suit for something a bit more vibrant. This became Spider-Man 2099's regular costume after its debut, but check out the Back to the Future, Shock book to see Miguel's story come to a dramatic conclusion.
Vintage Comic Book Suit
First appearance: Amazing Fantasy #15
Designed by Steve Ditko in 1962, the original Spider-Man costume was actually red and black, and the blue seen on the edges of the suit were meant to be highlights. However, since printing techniques got better and brighter colors resonated better with readers, Spidey's costume changed in 1966. Any Spider-Man comic in the 4-year interim will feature this design, but do yourself a favor and read Amazing Fantasy #15 if you haven't already.
Last Stand Suit
First appearance: Amazing Spider-Man Vol. 2 (2003)
We all sometimes wonder what would become of our heroes if they'd taken a different path. In the three-part Happy Birthday arc, we find ourselves face to face with an older, more jaded Peter Parker, on the run from the NYPD. His story is a brief, tragic one, but well worth reading.
Homemade Suit
First appearance: N/A
Like the Stark Suit, this outfit first popped up in Captain America: Civil War, with a much better look at it in Spider-Man: Homecoming. It takes a few visual cues from the Scarlet Spider outfit but is ultimately its own beast. There's not a direct parallel in the main Marvel comics, but if you want to see Parker slinging webs in his early days, try the Spidey series that started in 2015.
Dark Suit
First appearance: N/A
Another suit designed for the PS4 game, the Dark Suit doesn't come from the pages of a Marvel comic. That said, it's not too far of a stretch from the costume worn by Blood Spider, a Spider-Man imitator and mercenary who first appeared in Amazing Spider-Man #367. That's not a particularly great story, so try the Return to the Spider-Verse arc from the Ultimate Spider-Man Animated Series, where Blood Spider is Peter Parker, but also a vampire.
