Mesosphere is the commercial entity providing services and product enhancements around opensource Apache Mesos. Apache Mesos is a solution for orchestrating containers for a grid computing like use case where multiple servers can be combined for executing a large job. They do this by dividing the large job in small jobs that can execute in smaller servers. Then these smaller job outputs are combined to create the larger job outcome. In Mesos, these small job execution environments are created on containers. These containers and the jobs that run on them are orchestrated through Mesos.
Taking cue from the fundamental concepts of that Mesos architecture, Mesosphere came out with a term called DC/OS which alludes to the fact that Mesos can take all servers in the datacenter and combine their hardware under Mesos to create a pull of resources for containers and jobs running on them, just like an operating system combines all hardware of a system to run applications on them. DC/OS is therefore Mesos’s one enhanced version to feel and behave like an Operating System for the entire Datacenter. To complete this idea, DC/OS also comes with its own package manager etc.