no-break space
A no-break space looks like any other space, but the line is never allowed to end on it.
In more detail
Text is cut into lines at the spaces, and any space will do — which is how a page ends up with a number at one end of a line and its unit at the start of the next, or somebody's title left dangling with their name below it.
A no-break space is the same white gap to the reader and a locked door to the machine setting the line. Where it stands, the two words on either side of it travel down to the next line together rather than being parted.
What it does on a real document
An invoice states a total of twelve thousand five hundred euros, and French writes that number with a gap inside it. Break the line in that gap and the reader meets twelve at one end and five hundred euros at the other, and has to read the amount twice. Held by a no-break space, it goes down whole.
French asks for one more thing: a gap in front of a question mark, an exclamation mark, a colon and a semicolon, and inside its quotation marks. Every one of those is a no-break space, because a line ending on one would leave the punctuation stranded at the head of the next.
Where it comes from, and why
The rule is older than any machine. A compositor setting metal type simply did not put a break where the sense of the phrase forbade one: the eye reads a group of words as a single thing, and cutting it in two costs the reader a moment.
Once a machine was doing the cutting, the distinction had to be written down, and a character was set aside for a space that binds. Every typesetting system since has had one, and a document carrying the character carries the rule along with it.