To get the notification related to your build you have launched the Team Foundation Build Notification tool to have a small popup alert every time a build finishes and starts.
But now you would like to go a bit further and receive a mail with the build result every time a build fails (or let's say do not succeed).
Use the TFS' alerts
The Team Foundation Server includes a powerful alert system, but by default, you cannot do so many things.
Use the Team Foundation Server Power Tools
If the Project Alerts does not allow so many things, the underlying alert system is very powerful. And you can use the Team Foundation Server Power Tools that allow a very fine control of the alerts.
You can then create a new rule like the following one :
Note that for the Send To field, you can either indicate an account name, or an email address. In the case of an account name, this corresponds to an AD account, and you must have configured the email address of that person in the Active Directory.
Concept taken from: http://www.pedautreppe.com/post/How-to-add-an-alert-on-TFS-when-a-build-does-not-succeed-.aspx
~jawed