In Private Cloud, you can use the scalability and resiliency of the public cloud, with greater level of control and security in a single-tenant dedicated environment. Private Cloud can be hosted on your own premises hardware or on a service-provider data center where you have your own dedicated racks and hardware.
An environment where you combine private and public clouds, such as OpenStack, AWS or Azure and a single-tenant dedicated environment . You will get the benefit of the 'pay-as-you-go' mode of public cloud, the security and granular control of the private cloud for business-critical applications and then the ultra fast performance and reliability of bare-metal dedicated machines.
Multi-Cloud is an environment which relies upon multiple public or private cloud vendors such as Amazon Web services, Google Cloud Platform, Microsoft Azure or Openstack. A typical scenario is where your application needs large pools of compute resources from private cloud and at the same time, it needs to quickly scale on a public cloud as per demand. This environment avoids vendor lock-ins creating redundancy and helps to run workloads on different platforms where it performs best.