What is a "card run"?

I assumed "card run" meant people farming a raid to earn the boss's card, but I tried joining an Argos party and they didn't start because I didn't "Use my card", so now I have no idea what "card run" is referring to... Am I supposed to be using a specific card set to make it go faster, or is there a different "card" item that gives you special rewards from Argos?