I was watching this video.
When the woman gets the card wrong, she says "Mist". Is this just the German way of saying "dang" or "crap"? Does it have bad connotation? Is it appropriate to use?
The literal translation would be "manure".
It is not only limited to excrements of cows, but describes any form of animal excrements mixed with some sort of mulch (mostly Straw) that can be used as fertilizer.
Usage is, as suggested by @TheBlastOne, similar to crap in English.
It is not really an offensive word. You would usually use it if something went wrong or you made an error. Even if it is a trivial matter.
Whereas to use "Scheisse" (shit) which is more offensive the mistake would have to be more serious.
Not quite serious, but Homer Simpson´s "d'oh!" comes close. See http://en.wikipedia.org/wiki/D%27oh, and note Home did NOT invent it.
"Crap" is close, too.
One might say "shoot" in English to avoid facies as one says "Mist" instead of "Scheisse" in German. "Mist" is what cows produce, sometimes mixed with the hay that lies on the ground in the barn. So it is not pure excrement and not really as bad, one might think.
The german word "Mist" is "dung" as well as "rubbish". A "Mistkübel" is a dustbin or a garbage can. A "Misthaufen" is a dungheap.
When ever someone speaking english uses rude words from a sexual context (most famous is "fuck") you would use in german a dirty word that comes from a fecal context (most famous is "Scheiße" - engl.: "shit").
The english word "to fuck" (in the meaning of having sex) is in german "ficken" and both come from the same origin "fuggen" which means "to fit into something" (also english "to fit" and german "einfügen" and "Fuge" come from that). But never ever a german-speaking person would say "ficken" when a english-speaker in the same situation would say "fuck". In German you say "Scheiße" or, which sounds less rude, "Mist".
Almost everyone uses dirty words in daily life. But it is really hard to learn them. You don't learn them in school nor in any official course. Everybody teaching a foreign language does as if nobody ever would use dirty words in other languages, but everybody does, in every living and every dead language.
"Mist" is the shit of a herbivore or omnivore, mostly a cow or a hog, mixed with a binding agent, mostly straw. The feces of a dog or an eagle are not "Mist" (but "Hundekot" and "Vogeldreck").
The exclamation "Mist!" is used like "crap!" in English. As Lukas said, it is not as offensive as "Scheiße". It could be used in children's tv without causing outrage.