What is it and why is it used/useful?
A sanity test isn't limited in any way to the context of programming or software engineering. A sanity test is just a casual term to mean that you're testing/confirming/validating something that should follow very clear and simple logic. It's asking someone else to confirm that you are not insane and that what seems to make sense to you also makes sense to them... or did you down way too many energy drinks in the last 4 hours to maintain sanity?
If you're bashing your head on the wall completely at a loss as to why something very simple isn't working... you would ask someone to do a quick sanity test for you. Have them make sure you didn't overlook that semicolon at the end of your for loop the last 15 times you looked it over. Extremely simple example, really shouldn't happen, but sometimes you're too close to something to step back and see the whole. A different perspective sometimes helps to make sure you're not completely insane.