You must arrest 10 criminals.
Criminals are the enemies that have their own life bar. One will always appear at the end of the Last Encounter (Wave 5), the only other time they will appear is during a random challenge while playing. In the ladder case, you must kill/ arrest him before killing all the other enemies in the wave or he will simply disappear. Arresting enemies also yields you more Gold pieces for unlocking the final level, then killing them does.
To arrest a criminal you must first deplete their life bar, at this point they will hunch over and you must either kill or arrest them. To arrest you will have to stay very close to the enemy, an Arrest bar will then appear, once this bar fills you will arrest the criminal.
During this time, melee enemies will constantly spawn, but just stay close and melee them and you should have no problem. If playing with a co-op partner the Arrest bar will fill much faster if both of you stand close to the criminal. There is also a consumable item you can purchase to increase how fast the meter fills, but its unnecessary.