I hadn't heard the term dogfooding used much before I started here, but it has already been explained so take a look here. The basic idea is that if you're not happy using your product (i.e. eating your own dogfood) then why should you expect your customers to be? Working at Microsoft gives you incredible scope to dogfood a wide variety of products.
As a Microsoft employee, I should be using Internet Explorer, Vista, Office, etc etc and I am. This doesn't necessarily mean I shouldn't run alternative products as well or when a Microsoft product doesn't provide the functionality I need.
As a Microsoft developer, I should be using Team Foundation Server for bug tracking and source control. I should be developing Visual Studio using Visual Studio. I should be profiling my code using VSTS profiling tools. Fortunately, I am, although not exclusively and probably not in some other parts of the company.
The main reason I think this is a good idea is because we get to feel any of the pain that customers do. We have extra incentive to fix any problems instead of ignoring them. We often catch problems early on before customers even see them.
I'll admit it, the process can be painful. The pain typically increases as you get closer to the bleeding edge of technology. For example, my Visual Studio dogfooding experience involves running the latest build of VSTS while developing. There are issues which delay my development, but facing these issues every day helps me drive improvements to the product. Imagine if your source control system went down - you'd want it fixed pretty quickly and that's just what we want from our TFS dogfood server.
Here's a few of things that I think need to happen for successful dogfooding:
If you're an application developer, are you using your own alpha/beta software before it is released to the public?