I wanted to know when you are given a sedative or tranquilizer, do you fall asleep or do you black out?

Hopefully neither: Sedatives and tranquilizer a should be used for their calming effects. Occasionally sedatives might be used to induce sleep. An overdose would result in blacking out.

Sleep or black out: Falling asleep and blacking out are two descriptive terms of the same general thing. Blacking out is generally used when referring to a sudden abrupt loss of consciousness and could take place from large sedative or tranquilizer dose. The same medication given in a lower dose could cause a person to fall asleep. Sleep referring to a less abrupt and more arousable loss of consiousness.