Tongue-in-cheek describes an exaggerated facial gesture — pushing a cheek out with one's tongue — meant to convey contempt or irony.
It may have originated as a theatrical device, a technique for an actor to indicate that a statement was meant to be sarcastic or a joke.
These examples show how the expression is used specifically to signify joking or sarcasm:
Amid a torrent of criticism, Appleton apologized, claiming that his comment was "clearly tongue in cheek" and taken out of context. (Washington Post)
"I wish him good health," Mr. Putin said, going on to add that "I am saying this without irony or tongue in cheek." (Washington Times)
Tongue and cheek is a mishearing of tongue-in-cheek. If you say this phrase quickly, you can hear how easily "in" can be mistaken for "and." But tongue and cheek doesn't evoke the mocking, comical image of someone making a funny face — it's just listing parts of a face.
The examples below show how common this error is:
There are a lot of people calling themselves witches in a political way, in a tongue and cheek way. (New York Times)
"The comments are a little tongue and cheek," said Visit Santa Cruz film Commissioner Christina Glynn, who is encouraged by what the movie has done for the area, both financially and culturally. (Fox News)
When you know the origin of the correct phrase, tongue and cheek makes less sense. It can be hard to convey attitudes like sarcasm in written language. When talking, we can alter our voice and make facial expressions that add additional layers of meaning to what we're saying, even contradicting the words themselves. Written words lack such external, physical clues to the author's tone, but expressions like tongue-in-cheek allow us to evoke some of those nuances on the page. Seriously!