A board represents the processes in your Company, such as Development, Sales & Marketing, HR and so on.
A project unites boards related to one product.
For example, you are developing a travel application. Your Company consists of 4 departments: Design, Development, QA & Support, Marketing & Sales. Your Project is called Travel App and you have the following boards: Travel App Web Development, Travel App Mobile Development, Travel App Product Ideas, Travel App Marketing Strategies and so on. The main thing is, all the boards belong to a single product = project.