Zero Day exploits generally refer to a security hole in some software which someone has found and announced (or leaked, as the case may be) to the world, but the software developers don’t know about it yet or otherwise haven’t addressed the issue. They have literally had zero days to fix it before it becomes widely known.
The preferred course of action is that when a bug or exploit is discovered, the person or group who discovered it discreetly gets in touch with the developer, describes what they found and how it can be exploited, and gives the developer time to release a fix.