Tactfully is the adverb: doing or saying something in a way that avoids upsetting or embarrassing the other person. You put something tactfully, tactfully suggest, tactfully point out, tactfully decline, tactfully avoid a subject, tactfully change the subject.

The adverb is where this word family is most useful in everyday English, because it modifies the act of speaking, which is where tact actually lives. Two of the patterns are worth learning as whole phrases. 'To put it tactfully' — as in 'how can I put this tactfully?' — is what people say aloud when they are searching for a kind way to deliver bad news, and the phrase itself warns the listener that something uncomfortable is coming. The other is 'tactfully + a verb of avoiding': tactfully ignored, tactfully overlooked, tactfully said nothing. Here the tact is in what was NOT done, and English uses the adverb to make that restraint visible and deliberate rather than accidental.

It is a neutral word, at home in speech and writing alike, and very common in workplace and advice contexts. The opposite is tactlessly, which describes blundering into the sore spot without thinking.