Project Zero is a project by Google, started on July 15, 2014 to make the web safer.
As we all know, using the internet has a lot of risks. Hackers can take advantage of bugs in your computer software and exploit various vulnerabilities to steal data and do other harm to you and your data.
Google does its own research to make the Google products safer and eliminate bugs. During this research, Google extensively test other products like Windows, iOS, Linux etc. Now the company has formed a separate research team to identify such vulnerabilities in various third party software and report it to the vendor. Bugs are reported to the vendor directly and will be published only after the vendor fix the bug or fail to fix the problem in a reasonable amount of time.
This whole initiative of Google is called "Project Zero".