A message tape was a microtape that had been recorded with a message for someone else.
As of 2268, Captain James T. Kirk had recorded a message tape for Spock and Leonard McCoy to listen to in the event of his death. Upon Spock's having effectively declared Kirk dead aboard the USS Defiant, McCoy informed the Vulcan that since he was about to take command, they should review the tape first. The contents of the tape were thus:
Bones, Spock. since you are playing this tape, we will assume that I am dead, that the tactical situation is critical, and both of you are locked in mortal combat. It means, Spock, that you have control of the ship and are probably making the most difficult decisions of your career. I can offer only one small piece of advice, for whatever it's worth. Use every scrap of knowledge and logic you have to save the ship. But temper your judgment with intuitive insight. I believe you have those qualities, but if you can't find them in yourself, seek out McCoy. Ask his advice. And if you find it sound, take it. Bones, you've heard what I've just told Spock. Help him if you can. But remember he is the Captain. His decisions must be followed without question. You might find that he is capable of Human insight and Human error. They are most difficult to defend, but you will find that he is deserving of the same loyalty and confidence each of you have given me. Take care. (TOS: "The Tholian Web")