The military refers to the armed forces of Eldia. It was originally formed as a defense against the Titans on Paradis Island.
The military consists of three regiments: the Scout Regiment, originally tasked with venturing into Titan territory but now serves as the military's main expeditionary force; the Garrison Regiment, which patrols and maintains the Walls, keeping Titans out and acting as the first to engage the Titans if the gate is destroyed; and the Military Police Regiment, the top-ranking soldiers who serve as the personal guards for the king and maintain order within the Walls. Additionally, the military supports the Cadet Corps, the branch which oversees the training and education of recruits in order to join one of the aforementioned forces.
Story: By 851, using the Executioner from Hell, the military had exterminated the majority of the Titans on Paradis, clearing Wall Maria of Titans for the first time in six years. Following the Scout Regiment's investigation of the origins of the Titans is revealed to the public, the military changes their focus to defending Paradis Island from aggression from Marley and other nations. In the year 854, Eren Jaeger coerced the military into deploying the Scout Regiment to the city of Liberio, using the potential death of himself--as the current holder of the irreplaceable Founding Titan and Attack Titan--as a threat.
Trivia: According to Isayama, a trend within the Walls is to badmouth the various military units. Although cadets are given the choice to join one branch of the military, they are able to leave their selected branch to join a different one. This is seen when Floch Forster, Samuel Jackson, Daz and Marlo Freudenberg left their original branches (the Garrison Regiment and Military Police, respectively) and joined the Scout Regiment.
Source: Military (Anime), available under CC BY-SA 3.0.