For the most part, Pathfinder changed little about balance
The primary maxims of 3.5e remain true: magic dominates everything, the more and higher-level magic you have, the better off you are. Magic is both powerful and flexible, allowing magical classes to be strictly-superior to non-magical classes in many cases.
If anything, Pathfinder actually made balance worse: nerfs to combat feats, and the distinct lack of better combat feats, which 3.5e published in supplements and Pathfinder never did, hurt mundane characters dramatically. And spellcasters received new class features, some of them very powerful, plus powerful new spells (e.g. paragon surge, emergency force sphere). There were some scattered nerfs to some core spells, but many are unchanged from 3.5e and still overpowered.
That makes figuring tiers fairly straightforward
Roughly speaking, prepared spellcasters who get 9th-level spells are tier 1, spontaneous spellcasters who get 9th-level spells are tier 2, spellcasters who get 6th-level spells are tier 3, and classes that get little to no magic are tiers 4 and 5. That rule of thumb will work fairly well across Pathfinder.
Tier 1—Arcanist, Cleric, Druid, Shaman, Witch, Wizard
This is pretty straightforward: these classes get the most powerful spells at a given level, and they can change their spell load-out daily, allowing immense flexibility. Pretty much the definition of tier 1.
The druid is notable for the significant changes to wild shape—those definitely hurt the class relative to 3.5e. But ultimately, the druid’s spellcasting was always her best feature, and she still has it.
The arcanist has a strange “prepared spontaneous” spellcasting scheme, but it’s highly advantageous. The only downside to the arcanist is the one-level delay on spell levels, à la sorcerer. Exploits are also rather powerful. On even levels (as well as 1st and 19th), the arcanist is easily the most powerful class in the game; on odd levels, he is still substantially superior to the sorcerer. Easily tier 1.
The witch is much like the wizard, though her patron familiar is obnoxiously vulnerable compared to a wizard’s spellbook. Hexes are mostly meh, but there are enough good, even excellent ones, that they are distinct plus. So yes, no reason to demote her from the first approximation.
The last new class, the shaman, is kind of like a divine witch, and the spirit animal is a much lesser vulnerability than the patron familiar.
Note that it is possible to get sorcerer or oracle into tier 1. It basically involves pumping your spells known and taking advantage of options—particularly with paragon surge and mnemonic vestment—that allow you to change your spells known on a day-to-day basis.
Tier 2—Oracle, Psychic, Sorcerer, (chained) Summoner
The sorcerer is the quintessential tier-2 class, since it gets phenomenally-powerful spells, but is locked into a particular set. However, there are options that allow at least some sorcerers to “unlock” things and gain the flexibility usually found in tier-1 prepared casters. Sorcerers that take advantage of those options are more properly considered tier 1.
The oracle is Pathfinder’s divine version of the sorcerer, so it is also tier 2. And like the sorcerer, the oracle has options for becoming tier 1. It’s also a better class than 3.5e’s favored soul, even ignoring those options.
The psychic is the occult-magic version of the sorcerer, and there aren’t (yet?) occult analogues to the options that allow oracle and sorcerer to become tier 1, so the psychic is distinctly tier 2.
The summoner is an unusual entry here, because it only ever gets 6th-level spells. However, the eidolon is very powerful, and the summoner gets summon monster IX as a spell-like ability at the same time as full-casters gain 9th-level spells. On top of that, many of the summoner’s spells, despite being 5th-level and 6th-level for the summoner, are higher-level for other classes—allowing the summoner to have access to higher-level effects despite nominally only having 6th-level spells. It even allows a way to produce discounted magical items with those spells.
Tier 3—Alchemist, Bard, Hunter, Inquisitor, Investigator, Magus, Medium (with archmage or hierophant), Mesmerist, Occultist, Skald, Spiritualist, Warpriest, unchained Summoner, maybe Bloodrager and unchained Monk and/or Rogue
Pathfinder’s panoply of two-thirds casters is to its credit: these classes tend to be well-balanced, almost by default, because 6th-level spellcasting hits a nice sweet spot in the system. Only one of these classes comes from 3.5e, too.
That class, the bard, actually struggles quite a bit here, though. The changes to bardic music both make it weaker and also make it far more obnoxious to play (I will never understand Paizo’s obsession with round-by-round accounting, but they use it a lot). Furthermore, the 3.5e bard was a class that benefited immensely from supplements, which are unavailable in Pathfinder and replacements for which have not been published. You can make an argument for the Pathfinder bard actually being tier 4, sad as that is. It took the “master of none” trade-off for “jack of all trades” a bit too seriously.
The skald does a much better job at the bard’s schtick than the bard does, in my opinion. It’s still versatile, while having a little more weight to throw around. Even jacks of all trades should have a little specializing in this system.
The unchained summoner, unlike other unchained classes, is actually a nerf, and serves well to bring the summoner back to where it ought to be.
Alchemist, hunter, inquisitor, investigator, spiritualist, and warpriest are pretty straight-forward two-thirds casters with a specialty. They work well. The magus is like those too, but it works very well.
The occultist I want to shout-out specifically, just for being awesome and feeling more magical than just about any other class in the system, to me. Also, while its spells only go up to 6th-level, the occultist gets a lot of spells known. Not enough to somehow make up for being behind 9th-level spellcasters and vault it into tier 2, but the occultist does a very good job covering a lot of bases with spells alone.
Medium only gets 4th-level spells by default, but with the archmage or hierophant spirit, it gets spells up to 6th level. That allows it to be tier 3. Not having one of those spirits makes it tier 4. Which kind of defeats the purpose of the class in my opinion, since you’re strongly incentivized to just stick to those spirits and not use others (until the very highest levels), but oh well.
The unchained rogue is kind of an interesting case here: as a rogue, with lots of skills, the rogue is about as versatile as a purely-mundane class can be. And the unchained rogue is just about the best pure damage-dealer in the game. Those two combined make a strong argument for being tier 3, though you could argue that skills don’t do enough and the class is tier 4. I tend to favor 3 for it. Note also that the eldritch scoundrel archetype trades half of its sneak attack for 6th-level spellcasting—even with halved sneak attack, the unchained rogue can out-damage a lot of other classes, and then it gets spellcasting too, so that archetype at least is firmly in tier 3.
The unchained monk can make similar claims, but not as strongly. The rogue gains more skills, and is the better damage-dealer.
And the bloodrager I’m even less happy about. The class feels very close to a great class, but the basic “cast spell on beginning to rage” feature is unconscionably delayed until 11th, meaning 50% of levels (and thus well more then half the time actually spent playing a bloodrager, since so many games don’t play at high levels) kills the class. The class works vastly better if bloodrage can include a spell right from 1st (well, 4th), and there are no level limits beyond the actual spells the bloodrager has. Even then, the 4th-level spellcasting holds it back a lot. Basically, in almost every way, the bloodrager looks very poor in comparison to the warpriest. Some would argue for it being tier 3, but I’m much more inclined to put it at 4.
Tier 4—Barbarian, Brawler, Fighter, Kineticist (optimally), Medium (without archmage or hierophant), Ninja, Paladin, Ranger, Vigilante
The fighter, particularly with archetypes, does pretty well for itself in Pathfinder (or at least, would if not for feat nerfs), and so is more comfortably tier 4 than in 3.5e. Still, the sheer lack of good feats to take hurts the fighter a lot, and prevents it from really being stellar.
The barbarian (chained or not) is... hurt, relative to 3.5e, but then the 3.5e barbarian was very close to being a 2-level class, and the Pathfinder barbarian does better than that. Rage powers give a reason to stay in the class, while in 3.5e it was a matter of picking up rage and then doing something else. Still, while the barbarian was rarely used for more than a level or two in 3.5e, a level or two were used a lot in 3.5e. The Pathfinder barbarian has much less uniquely going for it. There are better damage-dealers (including the fighter), which means that you really have little reason to play the barbarian unless you really want to be a “barbarian.” And even then, a bloodrager or skald would be a better choice, or a refluffed alchemist (vivisectionist) or summoner (synthesist) do vastly better. Still, if you play one, the barbarian still can dish out the hurt pretty well.
The other two classes shared with 3.5e, paladin and ranger, are here largely because of their spellcasting. Of the two, though, the paladin is doing much better, with the vastly-improved smite evil and the partial reduction in MAD. With those changes, the paladin is probably at the top of tier 4, and has some claim to tier 3.
The brawler is probably the only other candidate here that really has much claim on possibly being tier 3. Being able to change some of your feats daily is a nice big boost to versatility, and a feature we should see more of in warrior classes. But unarmed combat is awkward, and ultimately Pathfinder feats are lack-luster—the class would work so much better in an environment with better feats (e.g. 3.PF).
The vigilante is a very weird class, and in practice really awkward to play. That said, it has excellent skills and quite-solid damage-dealing. In some ways, it’s a somewhat-lesser unchained rogue or unchained monk. In my experience, though, its awkwardness makes it difficult to leverage what it has and the class struggles to contribute as much as it should. Vigilante talents are strictly (and intentionally) superior to ninja tricks and rogue talents. The vigilante also gets several archetypes with 6th-level spellcasting: those, particularly the zealot, are very good, and make the vigilante tier 3.
The ninja is in a similar boat, though it’s much more comfortable where it is. Ninja tricks are pretty much strictly superior to rogue talents, but inferior to vigilante talents. It partially makes up for that by being less awkward to play.
The medium was already mentioned: without archmage or hierophant, they are stuck with their half-casting and that’s pretty mediocre. The other spirit benefits can be nice, but you really want them on some other class that can take advantage of them, which the medium largely can’t.
Finally, the kineticist. The kineticist is a problematic class. It more-or-less doesn’t work as described. However, if you ignore the description of the class, and follow certain rules, the kineticist has a solid claim on tier 4. See our Q&A on the subject.
Tier 5—Gunslinger, Kineticist (blaster), (chained) Monk, (chained) Rogue
Pathfinder firearms are awful, which means the class that focuses on them pretty much is too. The gunslinger spends most of its class features trying to overcome the problems with firearms, and while it does that—achieving significant damage numbers—other classes can do that and more, or at least do that with far less hassle.
The kineticist played as described—as a blaster—is extremely limited, having mediocre damage output which will quickly burn the kineticist out. It actually struggles to even reach tier 5, to be honest.
And the chained monk and rogue just have... little to no reason to play them, even if we ignore their unchained versions. Mobile mystical warrior can be done better by almost every class listed under tier 3, while ninja is pretty much a straight upgrade over rogue (and vigilante arguably a straight upgrade over that, but again, awkward to play). The qinggong monk and the eldritch scoundrel rogue archetypes do add spellcasting to these classes and therefore improve them to tier 4, possibly even to tier 3 for the eldritch scoundrel, but even with those there are still better options.
Untiered—Antipaladin, Cavalier, Samurai, Shifter, Slayer, Swashbuckler
Simply don’t have enough experience with them. The antipaladin is probably tier 4, and the rest are probably tier 5, but I cannot claim to be certain about them.
But the shifter bears special mention because it marks the only time Paizo has ever admitted that a class failed outright. That’s a kind of staggering statement, considering some of the problems some of the other classes have, and how much community feedback there has been complaining about those without any acknowledgement by Paizo. For the shifter to uniquely get overhauled and an apology made to the community suggests that it was very poorly off indeed. I am still not in a position to judge the post-errata shifter’s balance.