non-root user how to install docker?

Update to the yamenk's answer:
There is now an official rootless mode for Docker: Run the Docker daemon as a non-root user
Here's an explanation of how it works from one of Docker engineers: Experimenting with Rootless Docker

This shouldn't be possible as it would be a major security concern.

When docker is installed on a machine, users with docker access (not necessarily root) can start containers. In particular, they can start containers in priviliged mode, giving the container access to all host devices.

More importantly, A user with access to docker can mount directories owned exclusively by machine root. Since by default, a root user inside the container will have access to mounted root-owned directories inside the container, this will allow any Docker container started by a non-root user to access critical machine stuff.

Therefore, the sequence of having a non-root user install Docker and start containers should not be allowed as it can compromise the whole machine.

Check this explicit comment from one of the docker maintainers.