What it means
To save face means trying not to look foolish after something's gone a bit pear-shaped. You cover the embarrassment, protect your image, and act like you've still got a grip on things, even if everyone can see the wobble. People use it for excuses, quick spin, or any little social rescue mission that stops you looking fully mugged off.
Usage examples
"He mucked up the report and tried to blame the printer to save face, but everyone clocked it and got on with the meeting anyway."
"He made up a quick excuse to save face after forgetting her name."
"They offered a small refund, mostly to save face after the mix-up."
"She laughed it off and said she was testing us, just to save face after getting the answer completely wrong."
"The manager threw in a free drink to save face once the booking had gone sideways."
Where it comes from
This one comes from the idea of face meaning a person's public dignity or social standing. English picked that sense up through contact with Chinese in the nineteenth century. So if you save face, you're protecting your reputation after a slip instead of standing there looking properly exposed.