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There comes a time when everyone has to play a character with ability scores that suck. This is a simple guide with compiled information on how to bypass or reduce some of the penalties to maximize your character's playability.
First of all note that your attributes govern most of your abilities, but the most important are attack rolls, damage, saving throws, hit points and spellcasting. Most of the times the key attributes of these abilities are fixed; there are however various feats and class features that tweak them, letting you change the base attribute for a certain save or skill (e.g. insightful reflexes from complete adventurer makes you able to swap your dexterity with your intelligence modifier to determine your reflex saves). The best resource for this kind of abilities is the great LordKiwi's thread, x stat to y bonus.
Another important topic is the selection of your class. Surely you can't expect to play a character who suffers from MAD, like a paladin or a dual caster and be good at it. Classes that depend on one or two attributes to fill their role, like druid, cleric, wizard, beguiler, etc are probably your best bets. What really helps even further is when an important attribute of your class is also the base of a saving throw (like constitution and wisdom).
Class features is probably your best bet improving your attributes. That is because classes offers various ways of improving or changing your scores (wild shape, polymorph, metamorphosis, etc), especially your physical statistics. A great ally to low scores are alternative class features, replacing the abilities you don't need because you lack the attribute required (such as turn undead and really low charisma) or replacing abilities to reduce MAD.
Last but not least, the selection of race must be addressed. There are races that bypass some of the negative effects of bad attributes, such as undead characters and constitution, or incorporeal races and strength. Moreover an important mechanic to boost weak ability scores is the aging effects and there are ways to bypass them with spells and other abilities. Templates is probably the easiest way of boosting your weak areas and filling the gaps.

Attributes:

The question is, what can you really dump? As i see it, there are two ways to deal with this problem:
[LIST=a]

You have really bad rolls, but one particular attribute is average to high (12-16), in which case you put it in constitution.

You have really bad rolls, but you need your abilities to be more challenging. You put your best roll to your casting/main stat and you choose a race that has no constitution score, such as undead (e.g. Necropolitan Human).

Now you'll probably need to focus on your mental attributes and be some kind of spellcaster that depends on wisdom (e.g. druid, cleric, psychic warrior). Wisdom and constitution are keys to your most useful saves, so naturally they receive priority. Intelligence is all about skill points and usually arcane spelcasting, so it would be wise to keep it at third place. Charisma is also a nice attribute, but depends on your class features and abilities, so it can be actually dumped. Dexterity and then strength are of the least importance, because they are easily replaceable.
So to wrap it up, the stat importance is:

• Constitution or casting/main attribute and no constitution score.
  
• Wisdom.
  
• Intelligence.
  
• Charisma.
  
• Dexterity.
  
• Strength.
  

Of course the order of the last four attributes can be easily changed, according to your character's style.

Class:

Classes that depend on two or more attributes (constitution excluded) are right out. So naturally favored souls, spirit shamans, warmages and most martial classes are bad choices. Typically you can't go wrong with:

• Druid
  
• Wizard
  
• Cleric
  
• Psion
  
• Psychic Warrior
  
• Ardent
  
• Beguiler
  

Also, these classes focus usually on constitution and wisdom and they deserve special mention:

• Swordsage
  
• Dragonfire Adept
  
• Incarnate
  
• Totemist
  
• Dragon Shaman
  

Races:

It should be relatively easy to choose a race with the above attribute guidelines. Typically anything with a bonus to constitution is a fine choice, as long as it doesn't lose out on wisdom. Small intelligence hits should be ok if you aren't a skill type character and the rest are more or less dump attributes. Some obvious choices:

• Human
  
• Dwarf
  
• Gnome
  
• Mongrefolk
  
• Dream Dwarf
  
• Uldra (if LA buyoff is allowed)
  
• Anthropomorphic Bat
  

A very important mechanic, mentioned earlier, are the aging effects. See if you can play a venerable character of your race, receiving -6 to physical scores and +3 to your mental ones. Then you can go necropolitan to offset the constitution loss and use spells and class features to get alternate forms to boost your attributes back to normal ranges. Another way is through dragonwrought kobold, but kobolds are generally not very easy to play with bad ability scores. Also if you play a venerable kobold you'll need to find a way to immortality fast, as you'll have just a few years of life before your time is up.
Finally, if LA buyoff is allowed in your game, consider taking a template or a race with level adjustment.
Of all the templates, two are probably the most useful:

• Divine Minion: This gives you the option of completely dumping your strength and dexterity scores because of the fast wild shape option. The god selection gives you an edge, but of course limits your roleplaying capabilities. If you want to play a melee character, this is great: choose anhur as your deity and get to spend all your life in a lion form (as you can stay indefinitely in your animal form). If you want a skill-monkey or a caster form (to wear magic items and so on) Thoth is a great option, as it offers the Baboon, with 15 str, 14 dex and +1 natural armor. Moreover, the wild shape ability this template provides opens up possibilities for feats that augment the shapes you can take (such as the broken dragon wild shape).
  
• Necropolitan: Cheap, removes the need for a constitution score and it doesn't even cost LA. You can then work on effects to augment your undead state (such as the spellstitched template). Remember though that you are now undead and can be easily manipulated or destroyed by a clerics.
  

Other templates of particular importance are:

• Telthor: One of the few templates that grants you the great incorporeal subtype at a relatively low LA (+2). This means that you can completely shun strength. Unfortunately your type changes to fey, which means that it is not compatible with necropolitan, making you able to completely dump two ability scores. Telthor telepathy can probably be used with mindsight. The damage you get when outside the area you guard is a joke, with a little fast healing you can easily overcome it.
  
• Dragonborn: Low cost and great possibilities. It can grant you a con-based breath weapon and boosts your constitution while reducing your dexterity. The mind draconic aspect is just awesome.
  

this is not the theoritical boards, no pun pun tricks here. This is information for casual players that have trouble with their scores.

Warlock- the class that is independant of its ability scores
Artificer- rocks too hard to be held back by low scores
Bard- Aside from a few feats, inspire courage doesn't need much
Dragonshaman- like warlock, but con helps more and needs to be closer to the fray. Second best IMO

Everything you listed is SAD, not "I don't need stats"

Dictum Mortuum wrote:

Damn, i forgot about eberron. I'll include artificer. Warlocks are ok, too. Bards are a little attribute dependent, unless you want only to sing :D. Dragon Shamans are already listed.

What does SAD stands for :P? Seriously attribute dependent :P?

Single attribute dependant. That 18 is f***ing everything to those classes. What bard doesn't want to sing?

Necropolitan Venerable Human:

• Str: 8 - 6 = 2
  
• Dex: 8 - 6 = 2
  
• Con: -
  
• Int: 8 + 1 (1 point) + 3 age = 12
  
• Wis: 8 + 9 (7 points) + 3 age = 18
  
• Cha: 8 + 3 age = 11
  

Total points used: 10 (10 point buy !)

You'll need corrupted wild shape [SUP]Libris Mortis[/SUP] to make it work and it's completely unplayable at levels 1-5. Also note that you should avoid melee and focus more on casting/summoning.