Smoke tests are tests that are run to ensure that the most basic functionality of a program works. So not testing all the edge cases but instead making sure that the main things that the program should do, are working.
I'm reading "The Phoenix Project" at the moment and in that book they mention where smoke tests get their name from. I'd never thought about it - in fact, in my old team we used to mime smoking cigarettes whenever we talked about smoke tests. We knew that this had nothing to do with the etymology but still, it was fun. I always just thought of it as running some tests and making sure that nothing blows up, which would probably cause smoke.
Turns out that the name comes from hardware testing - specifically circuit boards. When a circuit board is powered on for the first time and starts to smoke, then that means that it's broken.