TFS 2010 - How do I create email alerts when anything is checked into TFS, Build's Fail/Succeed or Work Items are Assigned/Changed?

This is quite simple to do with the right tools installed on your development machine:

1) Ensure that you have an SMTP Server Configured for your TFS box - as described on MSDN at "How to: Configure SMTP Server and E-mail Notification Settings in the Services Web.Config File" 
2) Install the Visual Studio 2010 Team Foundation Server Power Tools from MSDN
3) Open up the Alert Explorer which installs a set of alert actions. The alert explorer is accessible from several different menus within Visual Studio 2010. See screenshots below:

From the Team Menu:

On TFS Work Items:

From the top level TFS Server Node:

On Branches or Folders within the TFS Source Control Explorer windows:

There are several predefined alerts that come with the Visual Studio TFS Power Tools. These are shown below:

The alert filters are quite flexible. You can modify and change these Alerts in the alter definition editor as shown below:

DDK