A quick word of warning, as noted in the original announcement this is definitely beta software with more than a few unimplemented features and rough edges. You probably don't want to install this on your production computer, nor try super-adventurous activities the night before your doctorate thesis or big work project are due.
First, the pre-requisites - installation requires a valid (and likely activated) Windows 10 installation running at least Build 14316. To get this build, you must be signed up for Microsoft's insider program using a Microsoft account. Then enable the developer mode (under Settings > Update & Security > For Developers > Developer Mode) and set the Insider Preview build ring to "Fast" (under Settings > Update & Security > Windows Update > Advanced Options > Get Insider Preview Builds > move the slider to the right.)
Once Build 14316 is installed, click the "Turn Windows features on or off" in the Programs & Features pane of Control Panel and select "Windows Subsystem for Linux (Beta)" After the feature installs, reboot your computer.

Log in again, then open a command prompt and type "bash" - follow the prompts and you'll end up in a full-bore bash prompt in mere minutes.
After installation, you'll have access to many of the default Ubuntu CLI functions including bash, awk, sed, grep, manual pages, etc. The environment also includes a semi-functional aptitude which can be used to install packages (including working dependency resolution) from the official Ubuntu repositories. A quick demo of installing git, reading the man pages for git and navigating the directory structure is shown below.
Note that although GUI applications (such as Gimp) can be downloaded and installed, the CLI cannot launch them as it does not (yet?) have a functional GTK/QT/X11 environment to use for display. No word yet on if/when this will be added, but in the mean time get out there and enjoy some *Nix On Windows!