Anything you do tactfully is done in a sensitive and thoughtful way. Ending a conversation tactfully means saying, "Well, I'd better go! It was great catching up!" instead of "I'm sick of talking to you."
Literally "done with tact," tactfully is the adverb to use when someone says or does just the right thing — especially in a difficult situation. Tact is this admirable quality of thoughtfulness and grace. You might smile tactfully when your mom asks how everyone likes her new haircut, or tactfully change the subject if someone brings up the subject of your uncle's divorce. If everyone speaks tactfully to each other, the holidays will go much more smoothly!