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. This is undermined somewhat by incarnum classes being somewhere between "lackluster" and "somehow a worse paladin." Incarnum, as a whole, is most effective at higher levels: the ability to replace magic items is more relevant, feats are more common, and the restrictions on essence/chakra investments more open.
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.
While Incarnum, like all of the other 3.5 subsystems, didn't carry over to Pathfinder, 3PP Publisher Dreamscarred Press and author Michael Sayre brought it back as Akashic Mysteries and gives it a near-east sort of feel. While the system itself is very similar, there are a few notable differences. The terminology is changed, in some instances significantly (chakras are now mapped to equipment slots such as Belt instead of Waist) and in others less so (such as using "essence" instead of essentia). Veils (soulmelds) no longer prevent you from equipping magic items, and many of the veils include abilities that Incarnum was lacking in, including AoE and healing options. The classes are also notably different.
- The Daevic is in many ways an akashic Paladin. Rather than casting holy magic though, daevics are influenced by a specific emotional passion (like a Daeva, geddit?), gaining certain benefits depending on what they focus on. Wrath daevics are damage dealers, while Desire daevics fight at range and have some enchantment spell-like abilities. Daevics 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 debuff class similar to a monk. While not focused on unarmed combat, they have the ability to inflict all sorts of penalties with essence-infused strikes. They also gain unique abilities based on their philosophy, essentially a class path akin to an oracle mystery.
- 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.
There's also three new akashic races that each have packets of abilities turning them into a different subspecies. The core three are gamla (camel), sobek (crocodile), and suqur (hawk), but the packages also allow for rhino, elephant, vulture, tiger, and other bestial humanoid types.
Dreamscarred also released a follow up class by author Anthony Cappel.
- The Rajah is a class that blends together Akashic and Path of War by being a party buffing 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. Extremely powerful in combat and pushes even the higher power levels of Path of War.
Lost Spheres Publishing also has several classes and options using the akashic subsystem and written by the original Akashic Mysteries author, taking the flavor in a planar direction and significantly refining and expanding the options available.
- The Eclipse is essentially an akashic shadow dancer. It has the lowest number of binds of any of the akashic classes but very powerful class features. It can create a shadow clone called an occultation and channel its attacks and veil powers through the occultation. It also gains the ability to wear its occultation as magic armor, turn its occultation into traps, or make and control multiple occultations. Starts with a darkvision feature that can be invested with essence to improve its functionality.
- The Nexus gains a warlock-like blast called a planar detonation and a set of class features called convergences that give it various abilities tied to a particular plane. The first two convergences are always a defensive ability, typically energy resistance, and a modification to the shape and damage type of their planar detonation. They get a full 10 veil binds but no unique bind, and their veil sets (thematic groupings of veils/soulmelds with built in synergies) are tied to planes like Heaven, the Hells, or the Underworld.
- The Radiant is an akashic healer that can invest essence into allies to give them buffs and then withdraw the essence to remove negative status effects. Very effective buffing and condition removal but reliant on its veils for its healing.
- The Zodiac is basically a summoner/binder/fighter hybrid that summons celestial beings called "constellations". Constellations come in armor, champion, equipment, and weapon forms. The armor and weapon forms of constellations are typically a suit of armor or specific weapon that gets automatic upgrades and can be enhanced by investing essence. Equipment forms are a unique magic item that can also be improved by investing essence, and champions are companion creatures (NPCs and animal companions) that can unsurprisingly be improved by investing essence. Zodiac also picks two orbits or paths that determine some of its class features. Solar zodiacs gain bonus feats and act as akashic fighters, while lunar zodiacs gain veilweaving and a unique list of support and control veils.
Lost Sphere's campaign setting, City of 7 Seraphs, also includes new veils and options for the original three classes (daevic, guru, and vizier).
- Daevics get the Knowledge passion, which gives them an "eidetic cache" they can store information or even spells in. Splits into Education and Secrecy.
- Gurus get the Shanti philosophy which maps their nonlethal motif up to 11. Capstone is an ability that turns weapons which strike the guru into non-magical tools.
- Viziers get the path of the scholar which gives them the ability to scribe and use scrolls, basic knowledge, and a capstone called Akashic Records that gives them a personal library demiplane containing every scroll or book they've ever read.