Racers are the vehicle you drive around from the original game. They are the weakest of the three vehicles you can choose from. Lancers are the APCs that the Crimson Lance drive around. They essentially look like a tank with wheels. The racers become available for you to drive after you talk with Moxxi for the first time.
Once you are able to drive a racer, head on up to "The Ridgeway." If you haven't already, grab a racer and head down the ramp. At the bottom there should be a Lancer. The easiest way to kill it is to stay as far away as you can and shoot rockets at it. If you are being shot at, simply swerve back and forth. The weapon on the Lancer is direct fire, so if you move around they can't hit you. Once the Lancer is destroyed, the achievement will pop.