A Freudian Slip is, as one wag has put it, "when you say one thing and mean your mother". It is an involuntary word substitution that supposedly reveals something you're repressing, hiding, or simply trying not to talk about. As such, it's a perfect tool for the comedy writer.
This is a specific form of faux pas which usually leads to Verbal Backspace; it generally comes as a result of being Distracted by the Sexy, as Freudian psychology often involves the primal desire for sex, but it isn't a strict requirement. Taken to extremes, it becomes a Freudian Slippery Slope. Contrast and compare (respectively): Last-Second Word Swap, Ignore the Disability. Super-Trope of Wrong-Name Outburst. Compare also Thought-Aversion Failure.
What the Freudian slip hides is called Shadow Archetype. Often these turn out to be Innuendoes or other sexual references, both for Rule of Funny and to give the characters an opportunity to observe that Freud Was Right.
Compare That Came Out Wrong and Accidental Innuendo, which are usually completely innocent and not noticed until someone says "Heh Heh, You Said "X"" or "That's What She Said". Contrast Lampshaded Double Entendre, which is quite intentional. Compare and contrast with the Fauxdian Slip, in which a character wants to openly express what the character making the Freudian slip is holding back on. See also Boggles the Mind, which plays mostly the same role in a different context.
See also Does This Remind You of Anything?, which includes subtexts that are only visible to the audience. Compare Have a Gay Old Time, when this occurs when a word's meaning changes over time. Likely to come about via Spoonerism where the first letters of subsequent words are switched. Compare You Just Told Me.
Suspiciously Specific Denial is totally not related, because it's not also about revealing more than you meant to say, or anything like that.
