There are two ways in which public and private services can be mixed: the more commonly recognized hybrid cloud, in which the private cloud resources and public cloud resources are merged in some fashion allowing service portability between the two, and a more pick-and-choose model in which individual services are assessed for feasibility in a private or public deployment model. Both have pros and cons, and the two can always be mixed. The key is ensuring the best business value for individual services without creating an environment that's too complex to manage or monitor.
The most commonly used example of a hybrid cloud is for the purpose of "cloud bursting." Cloud bursting is the concept of running your services on a private cloud architecture during normal operations and pushing excess capacity into the public cloud during peak periods. A great example of this would be a retail store whose compute capacity is fairly moderate and static from January through November but peaks significantly during the holiday shopping season. Cloud bursting would potentially allow the store to run in-house 10 months of the year and burst into public cloud on an as-needed basis with a pay-for-use model.
The immediate advantage to the concept of bursting is that you pay for only the predictable capacity that you can plan for, and you run that capacity at very high utilization with the safety blanket of the public cloud for both expected and unexpected peaks. The reality may be slightly different. The reality of cloud bursting is that it's not as easy as it may seem, and doesn't make sense for many models. The reason is that, if you've designed an application or service to run on a public cloud and you trust your data and security on that cloud and the public cloud's availability works for you, why are you running any portion of the service in-house?
While cloud-bursting use cases may be limited, other hybrid options exist. A great example of this would be utilizing cloud storage or other resources for business continuity and disaster recovery. The cost of maintaining a second data center for business continuity purposes is out of reach for most organizations; at the same time, the cost of losing their primary data center would most likely put them out of business. For companies that have chosen a private cloud model, public cloud services can provide a lower cost on demand pay-for-use disaster recovery strategy. While the public cloud may not be an ideal primary deployment model for that organization, it may be acceptable in disaster scenarios where the alternative is bankruptcy.