Ok, so this is pretty simple.
The public cloud is a product from a commercially viable company which provides everything from Architecture to compute to storage for a price. Some of these providers are Azure, AWS etc.... Public cloud is just that public. They are typically something that can be easily accessible from anywhere in the world if your on the net but are driven by the corporate giants. Equipment is not typically owned by the customer but leased. Security is not as secure as private clouds
Private cloud is that in which a company hosts its own data centers, their own infrastructure and applications. Privacy is paramount in private cloud. The private cloud typically owns all the resources and has to physically add / rack and stack new equipment as needed