Best Level Ever: Green Hill Zone. The first experience of Sonic for those who have been with him from the very beginning, it would stick into many fans' minds for the impressive scenery, the ability to just run forward, the crumbling terrain, and especially the moment where Sonic spins into a pipe and suddenly gets launched high up into the air. As such, it's an icon of the series that has been repeatedly homaged and sometimes outrightreappeared in future games. Apart from Green Hill, Starlight is also popular for being the other level to really showcase the speed.
Bootstrapped Theme: The title screen music (although it's rarely used in other series entries because of royalty issues with Dreams Come True, who composed it), and especially Green Hill Zone.
Dancing Bear: The game was sold on and became popular for two aspects. One, it was a blistering fast game for its time, allegedly thanks to the brazenly marketed "Blast Processing" abilities of the Sega Genesis. To a lesser extent, it was also for having a mild streak of edge to it that was lacking from the more vanilla Super Mario World and Sega's own previous Mario wannabe, Alex Kidd, while still having the lighthearted cartoon appeal lacking from Sega's other arcade ports and more mature games for their console.
Fan Nickname: The "Checker Wrecker" for the very first boss fight in the series, which was the name given for it in the Archie comics, but nowhere else. Plus, it sounds much catchier than the bosses canon name of Egg Mobile-H.
Fridge Brilliance: Why did Robotnik's machines in Sonic 1 seem so lackluster compared to the things he'd make later? Because Sonic just plain caught him off guard. It isn't until the last bit of Sonic 2 that he starts making machines specifically to counter Sonic. Until his future nemesis came along, a lot of his machines could easily trump most things that tried to get in the way.
Goddamned Bats: The "Roller" enemies in Spring Yard Zone, which move incredibly fast and appear quickly.
Good Bad Bugs: In the 2013 mobile version, it's possible to get Knuckles and Tails together via glitching.
The choral "SEGAAAAAA" at the opening of the game. It's reached Memetic levels due to how soothing and cool it sounds.
The opening bars to Labyrinth Zone may not seem this way at first but they're always the first thing you hear after the drowning music if you manage to save yourself from drowning. It sounds like a musical version of a sigh of relief. Similarly, the "wo-WA!" noise Sonic makes as he breathes the bubble is one of the most beloved sounds in gaming.
Nightmare Fuel: The infamous drowning music, which would become a recurring jingle in most games in the franchise with bodies of water that don't act as automatic Bottomless Pits. And since Labyrinth Zone is considered the hardest of the water levels in the series, you can expect to hear this a lot.
Also, the "Roller" enemies in Spring Yard Zone appear out of NOWHERE, but only if you try to go backwards. If you aren't expecting it, and you are playing late at night, in a dark room, close to the screen... Still gives this troper nightmares...
Polished Port: The 2013 iOS and Android re-release, contrast to the GameBoy Advance port, and it's handled by Christian "The Taxman" Whitehead, who worked on the enhanced re-release of Sonic CD before it, in collaboration with Simon "Stealth" Thomley. This version of the game features true widescreen support, the ability to play as Tails (and have him follow Sonic) and Knuckles, couple of other features new to this version of the game such as a Debug Mode that can give a variety of options other than bending the rules.
Porting Disaster: The Game Boy Advance port Sonic The Hedgehog Genesis, which features terrible and even glitchy physics (due to not using the original game code), butchered audio, and slowdown. To make matters worse, this massacre of a classic was released to mark the 15th anniversary of the series on the same day as Sonic the Hedgehog (2006), which had similar problems. The port of the game on the Sega Dreamcast's Sega Smash Pack: Volume 1 compilation fared better, but was also marred by butchered audio. The DS port included in Sonic Classic Collection is much better, smaller screen notwithstanding.
Scrappy Mechanic: The "spike bug" for those used to playing the sequels, which lack this behavior.
Signature Scene: Sonic popping up and finger wagging while smirking at the player in the title screen, and the entirety of Green Hill Zone, are the series most iconic elements by far.
Nightmare Fuel: Labyrinth Zone. Forget everything you know about the 16-bit version's bright, colourful textures and catchy music. This rendition is downright dark with creepy music to match, especially since it contrasts so heavily with both the previous zones (Green Hill, Bridge, Jungle) and subsequent ones (Scrap Brain, Sky Base). Worse yet, the boss is fought not only entirely underwater (slowing Sonic down) but also during the era in which you had no rings for boss fights.
YMMVs for both the Genesis and Master System/Game Gear game:
Anticlimax Boss: The final boss for both games. In the original games case, it is a fairly challenging, tedious final boss, and the lack of rings doesn't help, but it's not a particularly exciting battle—you're just locked in a room where Eggman tries to crush you to death with pistons, and then sends electric sparks flying after you. The game gear final boss is insultingly easy and pedestrian, where Eggman's weak spot is out in the open and he has no recovery time for his collision damage, with your only threat being your lack of rings, and an easy to dodge combo of floor-mounted flamethrowers and the occasional electric spark flying down at you—he can literally be beaten in less than 30 seconds. Fortunately, both games let you deliver a final blow to Eggman while he's escaping with just the right timing.