soft hyphen
A soft hyphen marks a place inside a word where it may be broken, and is drawn only if the break falls there.
In more detail
Set a narrow column and sooner or later a long word will not fit on the line it lands on. Either it goes down whole and leaves a gaping hole behind it, or it is split — and a split has to fall where the language allows one, not between any two letters.
The soft hyphen is how whoever writes the text says where those places are. It sits inside the word and is invisible; only when the line ends there does a hyphen appear on the page. Everywhere else the word is drawn whole, as though nothing had been marked.
What it does on a real document
A column eight centimetres wide in a leaflet, and a word of fifteen letters. With nowhere to break it, the line before it is stretched until the gaps between its words are wider than its letters: the page looks combed out, and the eye stumbles on every line.
With the places marked, the long word gives up its first half at the end of the line and the spacing goes back to normal. It is the same text either way. What changes is whether the reader notices the setting at all.
Where it comes from, and why
Words have been broken at the end of a line since the first printed books — the Gutenberg Bible is set that way throughout. The rules belong to the language and not to the printer, and English, French and German do not break a word in the same places.
When text moved onto machines, a character was set aside for the job so that a word could carry its own break points wherever it travelled. That is why the same text can be poured into a narrower column years later and still break correctly.
See it on a finished document
Soft hyphens and no-break spaces Break points after a slash or a hyphen