Dishonored: Dunwall City Trials features four distinct challenge types: Stealth, Combat, Mobility, and Puzzle. Of course, these are the four main pillars of gameplay in the original game; so these trials are designed to test your skills in each of these areas.
The Stealth challenges might ask you to eliminate a certain target in a house full of people. However, you must find the clues that identify the target before closing in on your quarry. The target changes each time, so you can't simply beeline for the woman in the blue maid's outfit. Another stealth challenge asks you to break into a house and rob the owner completely blind.
Combat challenges are fairly straightforward, though some of them feature special limitations. For example, Assassin's Run has you charging through a line of rooms filled with enemies armed only with a crossbow, and Oil Drop simply asks that you destroy the whale oil tanks as they fall.
Mobility challenges make heavy use of Corvo's Blink ability, but there's some variation here as well. One of these challenges simply requires that Corvo touches as many beams of red light as he can before the time runs out. Another has him racing to a specific goal. Yet another has Corvo racing to the bottom of a chasm while performing Drop Assassinations, which of course nullify falling damage.
Puzzle challenges are fast-paced, but require sharp thinking. One has Corvo using Bend Time to kill as many enemies as possible in one go, and another has him using environmental traps and special powers to build a chain of kills.