Basically, the factotum is a super-nerd who uses intelligence and observation to mimic other classes' abilities a couple times per encounter. This guy is quite possibly the best Skill Monkey in the game, has access to every skill in 3.5E including the Iajitsu Focus skill from Oriental Adventures, but is mostly known for breaking the Diplomacy skill. The class can also imitate the class features and spells of other classes, effectively making it the Harry Flashman of the party. One turn you're casting Greater Shout and other high power sorcerer spells, and then you're healing party members like a cleric.
If you're not cheesing, the class is squarely in Tier 3, with immense flexibility but no truly absurd powers. If you want to play a jack of all trades, this is indisputably the best class for that. It also works well with the Chameleon prestige class from Races of Destiny.
The typical class for people who can't make up their mind (and have even heard of it), and potentially the most indecisive badass of all time.
Broken down, the factotum is actually pretty bread-and-butter. You need Int, Con, and presuming you play up the skill monkey role (you should), you get Cha and Wis. You can't exactly "dump" Str and Dex, but you don't need any bonuses there, either. Of all the classes in 3.5, factotum is the one who actually benefits most from an "average" range in his stats: nothing really higher than about a 16 is actually necessary, but nothing lower than a 10 should ever be allowed.
What do you get for all this? You get average hp and BAB, decent proficiencies, and every skill in the game as class skill. ALL OF THEM. Not just Iaijitsu Focus (which is grade-A fucking cheese), but also Autohypnosis, Psicraft, and Use Psionic Device (which is... less cheesy, but still pretty stout). Oh, and while you don't get quite as many skill points per level as a rogue, your SAD for Int means you don't actually give a fuck, because you are going to have double-digit skill points every level in short order.
But wait! There's more!
The big mechanic for factotums is Inspiration Points. Basically, every "encounter", you get a pool of points to spend on your class abilities. Those abilities are boiled down to adding either your Int bonus or your factotum level to every fucking roll you will ever make. This is why you didn't need much Str or Dex: with enough Int and Inspiration, if you do happen to wander into combat by accident, you can hold your own for a hot minute until the party beefcake takes over. And with access to martial weapons, you are definitely not a slouch in that department; since you only get light armor/shields, you'd be silly not to go with a greatsword or some other weapon with delicious damage output. And if you think the Inspiration Point pool isn't big enough then this hard to find Font of Inspiration feat will provide it with a much needed boost, especially if you take it more than once.
And that's not all. You also get a bag of tricks that will basically make you useful in a dozen situations. Trapfinding, a few per-day arcane spells (which is literally your factotum faking knowing wizard magic with word salad and interpretive dance moves), a little lay on hands style healing (which is literally your factotum running through a bundle of holy symbols, looking for a deity that will take pity on your dumb ass to help you out), some sneak attack damage, extra actions, and the ability to ignore spell resistance and damage reduction.
Hold up there, chief. It's not all sunshine and rainbows. You have a hard cap on how much you can do this shit per combat/encounter. You can never truly replace a full caster, ever (you have to pop Inspiration for your arcane spells, and they are gone afterwards), nor can you replace an actual cleric or other healer (because you have a set pool of healing per day that will never beat a wand or regular spell-slot healing). Your sneak attack is pretty nice, but a rogue can always do it better and forever; when you run out of Inspiration, you're basically an average BAB beat-stick.
So what's good about all that? You can fill in for a party member in a pinch, always. You're not quite a fighter, but a few buffs can make you feel like one. You're not quite a rogue, but few good rolls and some luck will get you there. You can't really cast magic, but you can use items like wands and such (since you were smart and took UMD/UPD). Fact is, only a few classes with really niche abilities like bard, druid, and TOB silliness is the only things you can't really do.
You're basically the party Swiss Army Knife. You look a little silly, but every time someone says, "Man, I really wish we had a...", you're already doing what needed to be done there. If there are five party members and the basic four (fighter, cleric, rogue, wizard) are already done, it's a really tough call between a bard and a factotum as your fifth wheel character; both can be insanely useful depending on circumstances. Bard edges out slightly due to all the prestige classes that make it killer, but we're about to introduce you to some very fine 3.5 cheese...
Breaking Good (aka the Chameleon Build)
So in Races of Destiny, there's this little prestige class called Chameleon. The TL;DR version is, they were created by a master assassin who was tired of having to work so hard at faking being other classes to get close to a target. UMD is great in a dungeon, but trying to use wands and staves all the time when you're trying to pretend to be a cleric or wizard or something doesn't work forever. Someone's going to realize you aren't casting from memory and figure you out.
Mr. Assassin hit upon the perfect solution. He didn't have to do "everything" better than other classes, he just needed to "fake it until you make it" with them. So he visited a monk who was developing this ability to take on those abilities slightly, and change them day by day. Assassin murders the monk after learning everything he could, then goes home to create a whole school devoted to this new trick.
The Chameleon isn't correctly named, because you don't just appear to be another class, you ARE another class. You can only be human or doppleganger (or a changeling, ala Eberron, who is descended from both), and you need some pretty stiff skill requirements to get in (because few classes have all four of them as class skills). But once you meet those requirements, you get access to what is often considered a potentially broken class.
Chameleons can cast up to 6th-level spells from any arcane and/or divine list. Yes, any of them. This allows a chameleon with time, patience, and resources to cobble together a spellbook that looks like the ravings of a lunatic wizard, using cherry-picked spells from lists like bard and trapsmith to do shit like cast haste as a 1st-level spell, or even cure spells as arcane spells. The divine spell choices are slightly less good, because you have to pick a specific list for those spells; you can't mix cleric and druid, you have to pick one. Another slight drawback is that you have to meet normal spellcasting requirements, meaning you need at least a 16 in your casting stat, but since skill monkey usually already have such stats pretty high (party face requires Cha, all your scouting stuff needs Wis and/or Int, etc.), this is usually a non-issue for you. You also can do per-day class abilities like actual rage, paladin smites, even turning.
But the best part is, you can boost a single ability score all day long. Yeah, you can do this to shore up a bad stat you need to mimic. But when your chassis if factotum, this makes Intelligence the natural choice. You can jump in early with factotum 5, go full chameleon 10, and come back to finish off with factotum 10 (to get cunning surge, more Inspiration, and better SLAs and lay on hands).
Oh, also, at chameleon 2 you get a bonus feat that you can change from day to day, and at chameleon 7 you can get two "class" types at once (which means you can pseudo-theurge by taking divine and wizard casting at once).
There are drawbacks to chameleon. You can't use any of the abilities in that class, including spellcasting, to qualify for any-fucking-thing. No metamagic feats, nothing based on use of rage or smite, nothing. You CAN qualify for feats with any non-chameleon class, and your chameleon BAB, saves, etc. do still count for those sorts of requirements. But, for example, you can't take Arcane Strike because you have arcane spells as a chameleon; you only qualify for it if you get 3rd-level spells from elsewhere. (Factotum gets there at 8th level, FYI.) If you do qualify for Arcane Strike, however, you can use your chameleon arcane spell slots to fuel it; you just have to qualify for it elsewhere.
DMM cheese is similarly difficult to get without dipping cleric or something somewhere, because chameleon turning won't count to qualify for the feat... but once you do qualify, you CAN use chameleon turn attempts to fuel it.
As some of you really conniving fuckers may be guessing by now, yes, chameleon can lead to nine kinds of abuse. Shit, the free changing bonus feat per day alone is a reason some folks dip into chameleon anyway: wizards who don't want to blow feats on item creation, or who just want access to additional metamagic feats, may actually get more mileage from chameleon than loremaster or even archmage.
But for factotums, chameleon is basically a must-have. It's Factoum On Steroids, and it makes you infinitely better at your job. No, you don't get quite as much power as a Tier 1 or 0 caster, but Christ, you can still do a lot. Use action economy spells on the arcane side and cleric buff spells like divine power on the divine side, and you can truly lay waste to enemies a few times a day without breaking a sweat. You can save your wizard and cleric/druid from even needing to cast a spell for the first half of the dungeon, so when they get to the BBEG, they can just pop off everything they've got while you're backstabbing minions and worming your way up to the boss. Plus, if you were smart and took mental pinnacle, now you're also kind of a psion for a short time, and with UPD you can use all kinds of spiffy items like power stones and dorje.
Built properly, a factotum/chameleon is pretty much the perfect jack-of-all-trades character. In a party with a sublime chord bard, an eldritch theurge, and an STP erudite, a min-maxed chameleon will basically fit in so good that you won't need much else to conquer the world.