Which NASA mission used the first sky crane landing system on Mars?

The NASA mission that used the first sky crane landing system on Mars was the Mars Science Laboratory mission.

NASA launched Mars Science Laboratory in 2011, carrying the Curiosity rover. Curiosity landed inside Gale Crater on August 6, 2012, using a new entry, descent, and landing sequence. After atmospheric entry and parachute descent, a rocket-powered descent stage hovered above the surface and lowered the rover on a 7.6-meter bridle and tether system.

The sky crane was developed because Curiosity was too heavy for the airbag method used by Mars Pathfinder and the Mars Exploration Rovers. A conventional legged lander would also have required ramps and could have placed rocket engines close enough to the ground to threaten the rover with dust and debris. After touchdown, the system cut the cables and flew away before crashing a safe distance away.

Mars Pathfinder was an important predecessor, but it used airbags rather than a sky crane. The same basic sky-crane concept was later adapted for NASA's Mars 2020 mission and its Perseverance rover.