The Hybrid Cloud stands between the Public Cloud (which shares its resources with multi customers) and the Private Cloud (which restricts its resources to only a single customer), and often provides unique solutions that are just what you might require.
Let us say, you have an application that uses very sensitive data, and that the asset owner is unwilling to host the data in the Cloud. A solution would be to host the application and all other data in the Cloud, but retain the database containing the sensitive data "on premise", and create a Virtual Private Network (VPN) to the "on premise" asset to make it available to the application running in the Cloud. This configuration is represented in the schematic below.
Another situation could be that you have a Private Cloud, but it has limited resources, thereby impeding the upwards scalability of your applications. In that case, it might be advisable to create a Hybrid Cloud where your Private Cloud is paired with a Public Cloud, as in the schematic below.
What would actually happen is that if and when the Private Cloud ran out of resources, it could "borrow" additional resources from the Public Cloud. See the schematic below.
Many other configurations are possible, and the Hybrid Cloud just might be the best fit in any irregular situation.