This can be a bit perplexing. After all, no one has skin on their teeth. So what does it mean? It means you escaped or achieved something — death, a bad date, a top grade — by a very slim margin.
We have the Bible to thank for this phrase, and specifically the Book of Job. Job is a character who undergoes innumerable tragedies, and sighs, complains and rails against God because of this, although he never loses his faith. In Job 19:20, Job says, "I am nothing but skin and bones; I have escaped only by the skin of my teeth." He's saying he's narrowly escaped death — that he escaped death by a margin so slim, it's as thin as the skin on your teeth. No one has skin on their teeth; that's the point, and it's why it indicates such a minute amount [sources: Addis].