On Cloud, projects are split into two phases: Opportunities and Projects.
Opportunities are any job or project that you are trying to win. This includes anything from a lead to waiting for a signature to start a project.
Projects are won opportunities. You have received the signature and the deposit payment (or will soon) for the work you will do.
All projects come from an opportunity.
So, to create a project, be sure to create an opportunity and mark that opportunity as "Won." You won't be able to create a project without first creating it as an opportunity.
Once an opportunity is marked as won, it will be converted into a project and seen on the Projects page.