Incarnum is a Dungeons and Dragons 3.5e magic system that earned the title of "Most Confusing System Ever."
OK, so this is how it works: you have a bunch of mystical energy constructs called Soulmelds that sit where your magic items go, except you can't have more than one per item slot (which they call Chakras) unless you take a feat, and they don't stop you wearing magic items most of the time.
You then have two non-exclusive ways of charging these up: you can put points of magic gunk called Essentia in them to increase properties of your Soulmelds like DCs and bonuses, or you can bind them to a chakra (sometimes you can choose from a bunch of chakras for one soulmeld, unless you're a Totemist which has a special chakra that isn't actually a body part that turns the energy thing into a monster body part like claws), which does stop you wearing a magic item, but it becomes about as strong as a magic item. (At high levels of Incarnate, or other soulshaping classes if you take the appropriate feats, you can bind them to the heart and soul chakras, which replace the vest and robe/armor slots and are correspondingly more powerful.)
You can redistribute the invested essentia each turn between anything that uses essentia, granting you a great deal of versatility. That is, unless you take a feat that you can invest essentia into, in which case any essentia you apply is added all day (because fuck you, that's why). You are limited to the amount of essentia you can invest in each of these things by your Constitution modifier or an arbitrary number based on your level, whichever is lower, which keeps you from pumping your entire essentia pool into one ability and doing that over and over.
The purpose of the Essentia casting system was to make a unique magic system for D&D, unlike the standard Vancian magic, the spontaneous casting of sorcerers/bards, the at-will casting of Warlocks, the horribly broken skill-based casting of truenamers, the pact magic of binders, the item-based casting of artificers or the PP-based system of psions. The result is this mass (and mess) of rules. Once you get used to it, it's is not as unwieldy as it looks, and totemists are among the most thematically awesome classes in the game. "What's the scariest thing you ever ran into? Yeah, I can do that too."
Astute players might notice that most soulmelding classes can emulate magic items, making them potentially a decent user of Vow of Poverty. Very astute players will ignore this temptation regardless. Gestalts surprisingly well with Druid, since they can't have non-passive magic items anyways.
One thing that gets Incarnum noticed is that you can get surprisingly far into progression through feats alone, and pick up some interesting tricks for otherwise "mundane" characters with them. This makes incarnum feats worth a note in most optimization handbooks, especially for classes that get few features (Commoner, Fighter), or little support for them.
In Eberron, incarnum is native to Xen'drik, though they've spread to Shifters, who have learned a great deal about it. It's origin is unknown, but it is thought to be as old as "normal" magic and was either a: taught at the same time the dragons taught "traditional" magic b: A leftover of the Quori invasion that destroyed the giant empire c: it's a native giant tradition and/or d: the incarnum using races are descendants of the giants.
Another system that was thankfully left behind in the hacking that PF is, there's 3PP Publisher Dreamscarred Press who brought it back as Akashic Mysteries and gives it a near-east sort of feel. While the system itself is sorta identical (the most notable difference is how chakras are unlocked by level), the classes are anything but.
- The Daevic is a low-tier casting class along the lines of the Paladin. Rather than casting holy magic though, these guys are influenced by a certain emotional passion (like a Daeva, geddit?), gaining certain benefits depending on what they focus on. They also inherit the Totemist's not-body-slot through the Blood Chakra, which can manifest into a variety of special effects.
- The Guru is a mid-casting sorta-monk. While not focused on unarmed combat, they have the ability to inflict all sorts of penalties with essence-infused strikes.
- The Vizier is the Akashic equivalent to a Wizard, with a focus on using magic items as if he were arcane himself. His school focus, however, boils down to three ideals: Crafting, Intimidating, and Coordinating.
- The Rajah is a class that blends together Akashic and Path of War by being less of a combatant. They introduce a special discipline (which uses Essence to power it in the way Sleeping Goddess does with Psionic PP and lets you fire lasers with Strength) and a new slot called the Title, which you can bestow upon allies and buff them up.