The Titan Wars is a war between the Interstellar Manufacturing Corporation and the Frontier Militia that takes place on The Frontier within the universe of Titanfall. The entire story of Titanfall revolves around this brutal war.
Not a lot is known about the history of the conflict aside from what we see in the games, but it is implied that the Frontier Militia did not have access to functioning Titans in the first few years of the war, and relied heavily on Anti-Titan weapons such as the Mag Launcher to acheive victories. It is not entirely known how the Frontier Militia came into possession of Titans, but it can be assumed that they captured them from the IMC. It is also unknown if Milita Pilots are defected IMC Pilots, or if the Militia possesses the appropriate facilities to train new pilots.