In my experience, when looking for mechanical advantage, you simply can't go wrong with the 3.5 dwarf, unless you're playing a bard or sorcerer. Some people don't like dwarves for roleplaying reasons (although just as many, IMX, enjoy them immensely), but for pure power, they're on the broken side of ECL 0.
Ability Adjustments
Con is by far the most important stat in the current rules. Core only, Con applies to:
Hit points, Fort saves, barbarian rage duration, Concentration checks
Add in other non-core material and it can quickly be applied to spellcasting (Rokugan Earth Shugenja, and that's certainly dwarf-flavored, and an old broken feat) and AC (Races of Stone Deepwarden).
Best of all, Con applies to every single class. Spellcasters? Better hit points and Concentration checks. Rogues and monks? Better hit points and, for the former, fortitude saves against trap poisons. Front-liners? Better hit points.
Cha, although applicable to lots of different things using non-core material (it's actually the key for a 1-attribute character, in truly bizarre builds), requires a lot of workarounds to make valuable.
Key Features
+2 to saves against all magic and poison, coupled with that above-average Con and no penalty to a save-granting stat, means that a dwarf is likely to make the majority of the saving throws he'll ever face. Basically, he has to deal with disease and mechanical traps in the same way a normal character would. This ability would be worth a feat for each save, if it stacked with Iron Will, Lightning Reflexes and Great Fortitude - which it does. Compare it to the Cumbrous feat tree from Savage Species.
Darkvision is the best type of vision. It's not massive, but certainly it's very useful in the dungeon environment core D&D is designed for. Worth a feat, but not available through one.
No movement penalty in armor, aside from being ripped off directly from Warhammer, is a bit more limited - spellcasters and monks won't benefit from it. Nonetheless, it negates the major downside of dwarven speed for a cleric, fighter, barbarian or paladin. Almost worth a feat, but not available through one.
Secondary Features
Weapon familiarity, stability orc and goblin attack bonus, stonecutting, giant dodge bonus, two languages, a favored class of fighter (one of the best dipping core classes)... these aren't all that great individually, but they sure do add up.
Racial Classes
Dwarven Defender, Runecaster, Deepwarden... dwarves have the strongest racial classes by far. Dwarf Paragon isn't bad, either.
Overall
The main thing that makes dwarves so strong is that they have no real downside. Even as bards or sorcerers, they're still just slightly below average - the half-orc can't say the same.
A dwarf fighter or barbarian will generally break even in a duel with a half-orc, and defeat anyone else.
A dwarf rogue will generally perform about as well as a halfling or elf rogue.
A dwarf cleric will generally outperform any other.
A dwarf monk will generally outperform any other.
A dwarf paladin will break even, despite his Cha hit, because he'll have a net +1 save against most things and more hp to counteract the loss of some lay on hands.
A dwarf wizard will generally win a magic duel against a grey elf wizard, despite the latter's higher casting stat. And to even come close, I have to look to a non-core race.
A dwarf ranger will generally break even, being perhaps a bit behind a human.
A dwarf druid will be even better than every other core druid, although not by much.
A dwarf bard will be okay, but behind a gnome, halfling, human, half-elf or star elf.
A dwarf sorcerer will be okay, but behind a gnome, halfling, human, half-elf or star elf, possibly behind a regular elf, as well.
A dwarf paragon/wizard/runecaster will be leagues ahead of any other fighter/mage, wearing full plate and still lobbing excellent spells with NO chance of failure. He's almost on par with a non-dwarven cleric.
A dwarf fighter/dwarf paragon/deepwarden/dwarven defender will dominate every defensive battle. Only a large chain tripper fighter is better at defending a position.