The -v (or --volume) argument to docker run is for creating storage space inside a container that is separate from the rest of the container filesystem. There are two forms of the command.
When given a single argument, like -v /var/lib/mysql, this allocates space from Docker and mounts it at the given location. This is primarily a way of allocating storage from Docker that is distinct from your service container. For example, you may want to run a newer version of a database application, which involves tearing down your existing MySQL container and starting a new one. You want your data to survive this process, so you store it in a volume that can be accessed by your database container.
When given two arguments (host_path:container_path), like -v /data/mysql:/var/lib/mysql, this mounts the specified directory on the host inside the container at the specified path (and, to be accurate, this can also be used to expose host files inside the container; for example -v /etc/localtime:/etc/localtime would make /etc/localtime on the host available as /etc/localtime inside the container). This is a way of either feeding information into your container, or providing a way for your container to make files accessible to the host.
If a container has volumes available, either through the use of the -v command line argument or via the VOLUME directive in a Dockerfile, these volumes can be accessed from another container using the --volumes-from option:
docker run --volumes-from my_datastore ...
This will make any volumes defined in the source container available in the container you’re starting with --volumes-from.
This is discussed in more detail in the Docker Volumes documentation.