The system is only available right now for users who have build 14316 of Windows 10.
To get it, a PC has to be a part of the Windows Insider Program's Fast ring.
After installing the beta, users have to toggle Developer Mode on in:
Settings > Updates and Security > For Developers.
From there, they have to open up another settings tab. Check the “Windows Subsystem for Linux (Beta)“.
Restart the computer.
Open a DOS command prompt and run the bash command.
Windows 10 will then prompt them to confirm they want to install the Bash beta and run through an installation process before starting up the Linux subsystem with a Bash prompt.
The Bash prompt shows that it's being run as a superuser, but its permissions are still tied to the active user account unless it's launched using the “run as an administrator” option in Windows.
This may change in future.