In a series as complex as Attack on Titan, it only makes sense that the military force protecting the citizens would be equally so. Believing themselves to be the last remnants of mankind and trapped behind giant walls that keep out Titans, their only known enemy, it seems like there would be little need for a formal military. And while much of it is not bred for conventional warfare, these soldiers play a vital role nonetheless.
Humanity inside the Walls, eventually revealed to be members of the Eldian race, live inside the Walls on what is known Paradis Island. Eldia's military has three branches: the Garrison, the Military Police, and the Survey Corps. For the most part, these three branches function independently of one another with varying levels of competence. However, each has a function that's essential to the survival of Paradis Island.
The Garrison Protects the Walls and the People
Led by Dot Pixis, the Garrison is the largest branch of the Eldian military when judged by the number of troops. The Garrison holds overwhelming numbers because the bulk of new recruits each year opt to join this branch as opposed to the other two. Only the top 10 students per graduating class may join the Military Police, and most everyone else is too afraid of Titans to join the Survey Corps. The Garrison, however, is the branch that's most willing and able to work with the other two.
The Garrison is responsible for the safety and general wellbeing of not just the population, but the Walls themselves. Garrison troops on patrol are a common sight in Attack on Titan. If the Walls are being attacked, the Garrison will defend them, and if they're just dirty, the Garrison will clean them. As there has not been a Titan incident in the 100-year history of the Walls, most of this branch are lazy and inept, more likely to drink and play cards than hone their combat skills. However, they sustained the heaviest losses during the Battle of Trost. Despite their vast array of responsibilities, they are soldiers first and foremost.
The Military Police Maintains Law and Order
Under the command of Nile Dawk, the Military Police is meant to be the best of the best in Attack on Titan, comprised of only top recruits from each year's batch of trainees. While there is a Military Police presence throughout the Walls, they operate predominantly from Wall Sina -- the innermost Wall where only the wealthy and members of the government reside. All members of the Eldian military are trained to use 3D Maneuver Gear, but while the Survey Corp and Garrison's equipment uses blades, the Military Police uses guns. Far from the threat of Titans, the Military Police's primary purpose is to keep the people in line.
As a result of being the branch with the most power and furthest from the threat of Titans, the Military Police is corrupt almost to the core. They misappropriate taxes and freely take advantage of the underprivileged, even going as far as employing a former serial killer to carry out government-sanctioned assassinations of political enemies. The Military Police is viewed by trainees as the best life for a soldier as the path of least conflict, but the corruptive rot of the MP turns even the most skilled recruits lazy and self-interested.
The Survey Corps Searches for Answers and Fights for Freedom
Led by Erwin Smith, aka the GOAT, the Survey Corps are tasked with defeating Titans and claiming new territory outside the Walls in the name of humanity. Despite their noble purpose, the citizens of the Wall are generally not fans of the Survey Corps. Their expeditions outside the Wall cost a lot of taxpayer money and only ever end in mass death, with no progress made or secrets uncovered. That said, most in Attack on Titan consider the Survey Corps the best place to be -- for those with a death wish.
The Survey Corps is the smallest branch of the Eldian military, with the highest mortality rate and only a handful of new recruits per year. Their numbers dwindled as low as nine surviving members after the final Battle of Shiganshina against the Beast Titan, Reiner and Bertholdt. Despite its small numbers and poor reputation, it is comprised of the most disciplined and highly trained members in the entire military. Eldia would not have survived Marley's incursion without the Survey Corps.