As a hyphen-replacement, functioning as a semantic linking element in a compound word (most frequently: a compound noun) that binds together two elements more loosely than a normal-space-in-that-same-compound.For this parenthetical usage, the em-dash is more common in US typography, while the en-dash is more common in UK typography. This also works well with kerning (see above). The en-dash in this usage is always surrounded by spaces. As a modern punctuation mark for parenthetical phrases.The en-dash ( -) has two frequent usages: (Note that usage of either em- or en-dashes for parenthetical purposes includes many cases where you visually see only one em/en-dash, simply because of the implicit orthographic rule that the beginning or end of a sentence 'eats' one member of the 'dash-dash parenthesis symbol pair'.) Of course that would not apply to em-dashes surrounded by spaces (the NYTimes style guide recommends this, for this reason), but keep in mind that this makes for really fat typographic separation. While you could describe this as a question of visual taste, one definite problem with em-dashes without surrounding spaces is that tracking (aka letter-spacing aka 'stretching' in some TeX lingo) doesn't play well with this: interword spacing will increase uniformly for words separated by spaces but will remain fixed for words separated by an em-dash without surrounding spaces. (Some sources recommend 'hair spaces' around the em-dash for this practice, but other sources recommend against that, and I don't see this commonly done by newspapers and magazines in the US.) This practice is recommended against by some style guides these days (typographer Robert Bringhurst indirectly calls the em-dash 'Victorian'), and I agree. In US typography, parenthetical phrases are traditionally set off with em-dashes without surrounding spaces.The most common generally accepted possible use case is before a name in a quotation that is attributed only to a person (that is: not quoted bibliographically), say, at the beginning of a paper or chapter.The em-dash ( -) has very few recommended use cases nowadays.

(For example, I'm omitting discussion of the various uses of dashes in dialogues or quoted speech because (1) they pop up rarely in the (La)TeX world as they are mostly relevant to copyeditors of fiction (who can be assumed to have learned the rules otherwise) and (2) their usage is rare and thus not necessarily governed by hard-and-fast, rigid rules.) I am not saying that my answer is exhaustive. Note that I'm listing only the cases that occur most frequently and that seem to raise the most questions. Rather than hiding all the information in comments, here is a summary. The answers and comments I have seen so far are incomplete and not entirely accurate in some regards.

Long dash symbol