The Adeptus Titanicus (formally called the Adeptus Mechanicus Collegia Titanicus) is the organization which oversees the God-Machines of the Adeptus Mechanicus. These are used when a near-infinite amount of men, tanks, or Space Marines are still not enough. Generally in the service of the Imperium of Man, but Chaos has a few titans left over from the Horus Heresy.
The organization is structured into three halves: Maintenance, Training, and Fighting, with a secret fourth half of Construction, for a total of 200%, the tech-heresy of such claims being currently ignored by the Mechanicus, who should know better.
The Maintenance arm is exactly what you think it is, involving some of the most complicated and pants-on-head rituals ever created by the Mechanicus, which all actually do something valuable. Every part of the titan is considered both holy beyond comprehension and priceless beyond measure, so the best bet is to give the Adeptus Titanicus the smartest mechanics the Mechanicus can offer, and hope for the best. Which works all the time; again, these guys have their act together.
The Training arm draws titan pilots, called Princeps (which is not plural, the plural is principes [prin-ki-pays], which is Latin for "leader(s)"), from a training school called the Collegia Titanicus. It's like battle school from Ender's Game meets NERV from Evangelion, but the twist is that the pilots are not a bunch of whiny emo pussies but undergo training simulations to make them hard and manly. The Imperium is pretty stupid but they aren't that kind of stupid. You give a man a volcano cannon and he sure as hell better not mope cry and pussy around, but redirect that into glorifying the Emprah and RAGE against Xenos and the enemies of Man. Unlike practically all other armed forces, Princeps need to be able to think for themselves, be independent, and as such aren't brainwashed into a sludge and even have some real personality to them, as well as being the best of the best of the best. On top of all of THAT, they need to have a will strong enough to be able to properly interface with the mostly-sentient Titan without being subsumed by its gigantic consciousness.
The Fighting arm consists solely of keeping crew prepped and ready, refuel and rearm operations, and hauling the huge metal beasts back and forth between planets (which takes a lot of specialized hardware and custom cruisers to accomplish). It is always important that the Princeps believes in the Emperor that believes in him, because having an existential crisis while wired to the biggest killing machine in the Imperium is a bad scene all around. And, of course, playing intel jockeys for the titan detachments out in the field when it comes time for the "Legio to walk".
The Building arm is kept hush-hush, and operates under the guise of the Maintenance Arm due to similar needs in tools and resources. They commit borderline tech-heresy every day through assembling new titans, each of which are a unique machine in coding and construction. They actually must summon a functioning machine spirit from the main cogitators, and make it talk nicely to a human via neural interface ports. They don't build anything on the Imperator Titan scale, but they have made Warlords in the past, and are brought in whenever an Imperator or Emperor class needs serious repair. This isn't to say that the Mechanicus doesn't know the Adeptus Titanicus are making new titans, the Mechanicus are all for making more! The problem is that the methods used skirt the edge of tolerable doctrine (read: taking some design liberties), so the Mechanicus as a whole needs to look the other way.
Also the name of a table top game featuring Epic-scale Titans. But that game's sole playerbase consists of a single crusty neckbeard.