On a supported Mac, run:

brew install --cask docker
Then launch the Docker app. Click next. It will ask for privileged access. Confirm. A whale icon should appear in the top bar. Click it and wait for "Docker is running" to appear.

You should be able to run docker commands now:

docker ps
Because docker is a system-level package, you cannot install it using brew install, and must use --cask instead.

Note: This solution only works for Macs whose CPUs support virtualization, which may not include old Macs