queue: To put into an ordered arrangement, first-in, first-out. To queue something is to put it in a queue. An hour before graduation, people should queue up outside the gates to be let in.
cue: To start something, signal something to start, or configure something so that it's ready to start. When CJ says "Ladies and gentlemen, the President of the United States", that's Bartlet's cue to walk in and take the podium from her. She has cued him by giving him his cue. Or I'll cue up a record so that when I hit the CD player's start button at the end of the PSA I'm playing, the music starts immediately, skipping dead space or a pointless introduction.
Like "lose" and "loose", please G3T IT R1TE OR D13. kthx.