To earn the Protector of Gotham achievement/trophy in Gotham Knights, you must “Complete 50 Challenges”, which may sound like a fair bit of work, but in Gotham Knights “challenges” are both numerous and usually pretty easy to complete.
Challenges come from various sources, including training tutorials, special objectives for unlocking some [Momentum Abilities], busywork required to advance some main case files and villain case files and most importantly (as far as this achievement is concerned) various NPC challenges. To check what challenges you have available, just go to the “Challenges” tab in the Batcomputer, where you’ll find challenges sorted by category.
While all challenges count towards this achievement, regardless the category, to hit the magic number of fifty you’ll need to repeatedly complete NPC challenges. The sort of busywork you’ll be required to complete for various NPC challenges is random, but almost all of it involves you going around Gotham City and stopping opportunistic crimes and premeditated crimes. Sometimes you’re tasked with completing a set number of crimes, stopping specific crimes (Criminal Deals, Witness Under Attack, Bomb Threat, etc), rescuing civilians, or, in the case of Lucius Fox, crafting different types of gear or fusing modchips.