As the meta has developed, these tiers have changed.

The accepted answer gives a good idea of the general consensus of class power as of the time of the answer being posted, but that was almost a decade ago. More recent analysis has shifted things around a bit; a relevant forum post summarizing a large community project to re-rank the classes can be found at this link.

The method of analysis is mostly similar to what the currently accepted answer by Oblivious Sage mentions, though it is clarified a bit more. Specifically, the classes are rated in terms of what they can do on their own and not what they can bring into a multiclassed character build, and the analysis tends to focus on levels 4 to about 16, rather than trying to equalize every possible level of play. There is also significantly less focus on whether a class has access to game-breaking exploits.

The tiers are listed as such, with each class having a ranking according to the mean voted power level of the class:

Tier 1 (S)
Druid: 1
Sha’ir: 1
Cleric: 1.05
Wizard: 1.09
Archivist: 1.13
Shaman: 1.12
Wu Jen: 1.17
Artificer: 1.28
Spontaneous Druid: 1.31
Urban Druid: 1.36
Death Master: 1.49
Tier 2 (A)
Generic Spellcaster: 1.66
Spontaneous Cleric: 1.74
Erudite: 1.78
Psion: 1.78
Sorcerer: 1.81
Evangelist: 1.88
Beguiler: 2
Mystic: 2
Dread Necromancer: 2.15
Warmage: 2.2
Ardent: 2.2
Favored Soul: 2.29
Spirit Shaman: 2.49
Mystic Ranger: 2.49
Tier 3 (B)
Wilder: 2.55
Shugenja: 2.84
Bard: 2.91
Trickster Spellthief: 2.95
Jester: 3.07
Totemist: 3.08
Swordsage: 3.12
Warlock: 3.20
Binder: 3.21
Crusader: 3.22
Warblade: 3.26
Dragonfire Adept: 3.28
Healer: 3.33
Psychic Warrior: 3.32
Duskblade: 3.34
Lurk: 3.4
Psychic Rogue: 3.4
Wild Shape Ranger: 3.47
Tier 4 (C)
Incarnate: 3.5
Factotum: 3.51
Wild Monk: 3.51
Rogue: 3.79
Shadowcaster: 3.82
Barbarian: 4
Generic Expert: 4
Generic Warrior: 4
Scout: 4.15
Spellthief: 4
Paladin: 4.19
Ranger: 4.28
Ninja: 4.33
Savant: 4.37
Fighter: 4.48
Marshal: 4.49
Tier 5 (D)
Adept: 4.54
Truenamer: 4.56
Sohei: 4.58
Hexblade: 4.65
Battle Dancer: 4.73
Divine Mind: 4.75
Monk: 4.78
Dragon Shaman: 4.81
Mountebank: 4.84
Samurai (OA): 4.85
Magewright: 4.94
Soulborn: 5.01
Knight: 5.02
Noble: 5.05
Swashbuckler: 5.07
Soulknife: 5.22
Samurai (CW): 5.26
Tier 6 (F)
Expert: 5.73
Aristocrat: 5.76
Warrior: 5.8
Commoner: 6

Some of the stand-out changes include the following:

• Healer was moved up to Tier 3. It's not clear why the Healer was ranked so low to begin with, and it certainly never had any business being lower than the Adept. At the levels where pure healing starts to fall off, the Healer gains access to a unicorn companion that can eventually be traded out for companions with potent divine spell lists of their own.
• Truenamer is ranked at the top of Tier 5, right next to the Adept and just a stone's throw away from the Fighter. While there are issues with the Truenamer, the class is significantly more playable than most give it credit for. Reasonable assumptions can be made about how the class works, and thus it can absolutely be ranked. That said, of course, the class is still has a glaring lack of good mid-level utterances.
• Having incredibly specific power is generally less favored. For instance, the Adept and Expert have both dropped a tier because they are one-trick ponies who quickly run out of things to do when push comes to shove. Classes like the Paladin and Beguiler, however, are ranked a tier higher than before because they can lean into a range of strengths and can avoid over-specializing fairly well.
• Similar to the above, Erudite drops a tier even with spell-to-power considered because most of the power in that variant comes from cheesing the way spells are converted. A lesser version of this effect can be seen in how the Cleric and Druid are both ranked (slightly) higher than the Wizard, where the original list is ambiguous about their relative power levels and seems to imply that the Wizard is the strongest among them.