Lets start by looking at what the definition of Hybrid Cloud is. This article from searchcloudcomputing certainly sums up most peoples view.
A hybrid cloud is a composition of at least one private cloud and at least one public cloud. A hybrid cloud is typically offered in one of two ways: a vendor has a private cloud and forms a partnership with a public cloud provider, or a public cloud provider forms a partnership with a vendor that provides private cloud platforms.
A good way of describing Hybrid cloud is explained here from hostingnews.com
In some situations, the customer may need both a local server running specific applications and a cloud service that host’s additional applications, files, or databases. In such a situation, the two are often configured for interoperability.
Ultimately a hybrid cloud is not made up of just two separate environments, with isolated from each other, the two environments have connectivity with each via the use of a VPN.
Another way of looking at Hybrid Cloud is providing a mechanism that sits above all the different technologies that can manage and deploy to these entities. A service broker is a nice term that has been used in the past. A consumer comes along, requests the deployment of the workload, and it is deployed on the best platform or technology suited to that workload, as shown in the diagram below.
 
Hybrid-1
So why are customers looking at deploying Hybrid Clouds and what are the key business drivers?
· Reduced Total Cost of Ownership (TCO)
· Lower Capital Expenditure (CAPEX) when trying new markets
· Business Agility
· It provides a clear use case for public cloud computing
· Using a hybrid cloud model is a valuable approach to architecture
· Ability to mix and match resources between infrastructure availability
· Solve resource constraint headaches by scaling out to the cloud
· Innovation and new product development
· Accelerating release cycles and speed to market
o Faster Application deployment
o Prolong legacy applications