The first command using docker run
is to start a new container. Docker containers can be run anywhere - on your local machine, within a VM (Virtualbox, VMWare etc), in an instance in the cloud, on bare metal or even on your smartphone. All this requires is to have docker installed and running as a daemon / service
docker-machine
is a tool used to mimic running docker containers locally using a VM. This is only because earlier versions of docker were not available on MacOS & Windows natively. As such a Linux OS is made available insider a virtual machine with docker installed. On this VM it was possible to run docker commands and docker containers as though it was running docker natively.
You should check out Docker for Mac and Docker for Windows if these are compatible with your setup.