To talk carefully without hurting anyone’s feelings, that’s tact. Politicians have tact, which makes them good at speaking about sensitive matters without making fools of themselves. At least, sometimes they have tact.
Around a friend who’s afraid of snakes, you use tact when talking about reptiles because you don’t want to upset them. The Latin root word tangere means “touch,” and a person with tact avoids touching dangerous words like they are an electric fence. When you say something without tact, you “put your foot in your mouth,” as the phrase goes. You don’t literally put your foot in your mouth, although if you did you’d avoid offending people with words.