There are many ways of using Docker and obviously different companies could come up with their own business cases for adopting the technology. So let me focus on one scenario and we can talk about whether it makes sense for your environment.
Software engineering is often difficult because programmers have to deal with inconsistent environments for development and for production execution of their products. Due to mismatches between these environments, developers often found bugs that surface in one environment but not in another.
Hardware based virtualization (VMWare, HyperV, etc) helped with the inconsistency issue because it enabled developers to create dev/test environments that could later be replicated into production. However this category of virtualization requires more computational resources (esp. storage) than operating system virtualization like LXC used by Docker.
In addition to requiring less resources than hardware virtualization, Docker defined a convenient container specification format (Dockerfile) and a way to share these specifications (DockerHub). When used together, these Docker technologies accelerate the process of defining a consistent environment for both development and production in your application. Dockerfiles are easy to maintain and help reduce the need for a dedicated operations team for your application. In buzzword speak, your team can become more "DevOps".
Docker, by the virtue of relying on a thinner virtualization layer than hardware hypervisors like VMWare, also has higher performance for I/O intensive operations. When used on bare metal hardware, you going to be able to get better performance for many databases in Docker containers compared to databases running in a virtual machine guest.
So to recap, Docker can help you
- maintain consistent dev/test/prod environments
- use less resources than hardware virtualization
- free up the time your team spends on dev to ops handoffs
- improve your app I/O performance compared to running in a hardware virtualization based virtual machine guest
However if you are using AWS, note that Docker Container Service available from Amazon actually doesn't give you Docker on bare metal. That's because Docker Containers run in AWS virtual machine guests virtualized by Xen hypervisor. So with AWS you are paying a penalty in resources and performance when using Docker.