Stepping outside of the RBY OU competitive meta context, I agree that Psychic was the dominating type of the first generation. The cartridge game had poison types EVERYWHERE (Team Rocket loved them, Zubats flooded every cave, poison was paired with stupidly common Pokemon on routes like Venonat/Bellsprout/Oddish etc), it had no practical weaknesses, and the Psychic Pokemon tended to be gifted in terms of stats - plus Mewtwo. It was so clearly out of control that Game Freak had to nerf it like crazy in Gen II by introducing Dark. Game Freak did their best to "balance" in Gen I, but there were tons of oversights, and Psychic was definitely OP. So, Psychic in Gen I has a reputation that a lot of these posters are leaning on - as Isa mentioned, there are some names here we don't see play RBY competitively, but the general knowledge of "Psychic was OP in Gen I" is an earned reputation.
However, as Jellicent has pointed out, we're talking about the RBY OU metagame, not the cartridge game and not the Ubers tier. Psychic is still powerful, but it's not undeniably number one. Here's my list:
1) Normal. Everyone else has given their reasons, and I agree. Tauros, Snorlax, Chansey, Hyper Beam, Body Slam, etc. Golem/Rhydon resist it and Gengar is immune, but they all get coverage (except poor Persian v Gengar) - this means a normal type is almost always a good choice. I don't want to get too specifically into distinct Pokemon v Pokemon matchups, but there are very few things who WANT to switch into a normal type and eat an attack, whereas if you need to switch into an Alakazam, or Eggy, or Jynx, your team is more likely to carry something that can switch in more happily.
2) Psychic. This is a very close 2. I think we don't fear Psychic anymore BECAUSE we've accommodated to play around Psychic. Psychic resists Psychic so an Alakazam isn't really scary to another Alakazam, or an Eggy. Psychic has the POTENTIAL to be everywhere and so Poison types rarely make a showing (Gengar and Victreebel being the prime counterexamples, but they're riding on the coattails of their primary typing). It's a scary type - whether you use the actual move Psychic or not. But, Psychic rides a lot on that intimidation. We know Psychic is scary so we counter with other Psychics, we don't bring Poisons (even if that Slowbro doesn't carry Psychic), because of the intimidation factor. Normal gets the edge because it doesn't need to bluff to make its point.
3) Ground. Electric would be a terrifying type if it weren't for these guys, and Golem/Rhydon attack hard on the physical side. As for Ice v Ground, I know Isa threw the SPL usage stats up there, but tournament stats for just 4 weeks aren't too illustrative, especially in an extremely high-level tournament like SPL where people are building teams to specifically counter their opponents' play styles, rather than using a jack-of-all-trades team on the ladder, or against opponents they don't know well or haven't studied. I feel like Jynx has seen a huge resurgence as a lead lately, too, which means the stats up there could simply be representing a fad that could fade away. That being said, it's all my opinion and my opinion only. Personally, I think Ground is more mandatory on a team than an Ice, and Earthquake has more opportunities to be threatening than Ice Beam/Blizzard.
4) Ice. Of course, Ice is still huge, so it gets 4. Honestly, it's so close I wouldn't be surprised if Ice and Ground end up tying when this is all over. Ice Beam and Blizzard are fantastic, and the freeze chance can be devastating. As I said above, I think Earthquake has more opportunities to hurt. If you disregard the freeze chance, of the 16 Pokemon in OU, 6 clearly fear the Rock/Ground attacking combo more (Zam, Chansey, Cloyster, Gengar, Jynx, Lapras), 4 clearly fear an Ice attack more (Dragonite, Exeggutor, Golem, Rhydon), 2 can shrug off Ice attacks and take more damage from Ground, even if they aren't particularly threatened by Ground (Starmie, Slowbro), and the last 4 would fear them roughly equally depending on the matchup (Persian, Snorlax, Tauros, Zapdos). It's a crude analysis, but when it's this close it gives Ground the edge.
Those are my major thoughts. I have some general thoughts on ranking below that but I don't feel too terribly strongly.
5) Water. When I first read Jellicent's original post, my instinct was to put Water at 3. But after reading everyone's replies I agree that Water itself isn't really the powerhouse. There are tons of water TYPES, but they love their secondary typing more (Starmie, Slowbro, Lapras, Cloyster). Still, some very powerful, threatening Pokemon are Water-type (even ones like Kingler, who aren't technically in OU but can hold their own), and that can't be underestimated.
6) Electric. Electric would shred OU if it weren't for ground. They're fast with high crit rates and have good special stats. On non-Electric types, Thunderbolt is a staple on most of the Pokemon who learn it.
7) Grass. Double-edged sword here - Ground and Ice are pretty much tied for me, and Grass may resist one, but is weak to the other. Still, Eggy is arguably a mandatory addition to an RBY OU team, and the Grass typing helps it way more than it hurts it - the resistances to Electric, Water, and Ground are a more-than-fair tradeoff to the Ice weakness. As an attacking type, Grass can bring some hurt with Razor Leaf, and I personally have a soft spot for Leech Seed.
8) Ghost. I wish Gengar wasn't Poison, or this type would be so damn good. With only one Ghost Pokemon in the whole game, though, you can't ignore the fact that it's part Poison and therefore gets some devastating weakness to Psychic and Ground. Still, Gengar has a lot going for it (I've written a lot about Gengar previously) and that means Ghost still has a strong presence in the tier.
9) Rock. That normal resist is handy, and Rock Slide is a great move.
10) Flying. I wish more Flying types had a good STAB, but while Flying can often be more of a hindrance than a help on a Pokemon (Dragonite would love to ditch the typing, and weaknesses to Rock, Electric, and Ice is terrible), the Pokemon who happen to be Flying types can hold their own.
11) Dragon. Eh. It's kind of worthless as a type in RBY with only one Pokemon and no STAB and a terrible Ice weakness, but Dragonite can be a real threat anyway.
12) Poison. Poison typing brings nothing good to the table. It doesn't resist anything useful, its weaknesses are extremely unfortunate, there are no decent STAB attacks and even if there were they would only be useful against Eggy (Bel and Saur would be hit neutral) so everything that is half-Poison just wishes it wasn't. Still, there are viable half-Poison Pokemon in OU (Gengar, Victreebel) so you can't ignore Poison's presence.
13) Fire. Bad attacking type, inflicts a status you normally don't want to inflict. It has some "cool" Pokemon though that can be brought into an OU match. However, I see a potential for Fire: now that Body Slam can't paralyze Normals, is using Fire Blast to burn Snorlax and Tauros a not-terrible idea [EDIT to respond to Ortheore/clarify: as in, does finding a slot for Fire Blast (as in, in place of Body Slam on Goldon) seem more appealing for the burn chance? Probably not, but because you used to mostly hope for paraslam against those guys, Fire Blast just seems so interesting to me now]?
14) Bug. Pin Missile on Jolteon. Shrug.
15) Fighting. Fighting and Fire are the ONLY types listed with virtually no presence in OU. Bug and Poison are also terrible, but Jolteon brings Pin Missile, so Bug gets represented. Gengar and Bel are Poison-types, so Poison is represented. So why is Fire above Bug and Fighting? One, I feel like I see Ninetales, Flareon, and Moltres in OU far, far more often than I see Machamp, Poliwrath, or Hitmonlee. Maybe not Poliwrath. Two, I bumped Fire up a little bit because Burn would seriously hurt Snorlax and Tauros, even if it would be the worst thing that could happen to a Starmie or Chansey. Shrug.