From the project page
"Google Chrome to Phone Extension is a project consisting of a Chrome Extension, Android App, and supporting AppEngine server that enables users to send links from their Chrome desktop browser to their Android device using Android's Cloud to Device Messaging service."
The aim of this project is to help you bridge the gap between when you are on your desktop and your phone. With it, you can send links directly from your Chrome browser to your phone, place calls directly from your phone by highlighting a phone number on your desktop and populating the Android clipboard by highlighting text on your desktop.
To get this running on your phone, you'd need to download the freely available Android app and Chrome extension available here and follow the instructions here.
These kind of developments are very interesting in that they help point us to the direction the up and coming Chrome OS will be headed. It's a taste of how Google is aiming to bridge the gap between the desktop and the cloud through your phone. If you are an Android phone user, try out this project and tell us what you think.