There are three types of clouds − Public, Private, and Hybrid cloud.
In the case of public cloud, the 3rd party service providers make resources and services available to their customers via Internet on the public cloud. The data of the Customer and its related security rests with the service providers’ own infrastructure.
A private cloud also provides somewhat similar features as of public cloud given above, but the data and services are managed by the Company or by the third party only for the organization of the Customer. In Private cloud, major control rests over the infrastructure as a result of which the security related issues are minimized.
A hybrid cloud can be said the combination of both private and public cloud. The decision to run whether on private or public cloud usually depends on many different parameters like sensitivity of data and applications, industry certifications, required standards & regulations, etc. etc.