It's still around in Windows 7.
How does it work and how does it compare with newer sync options such as the native Sync Center?
Should we still use it if it's not being phased out?
To use a briefcase, you first create a briefcase file and put it onto your portable medium, such as a flash drive. Then you drag any file you want to sync into the briefcase. When you do this, it maintains a link back to the original file.
You bring the briefcase file to another computer and work on it from inside the briefcase. Then, when you open the briefcase on the original computer, you can sync the files so it updaters the local copy with whatever is in the briefcase (or update the briefcase if the local copy is newer).
Microsoft provides a somewhat dated explanation of how to use the briefcase feature. The briefcase seems to still be a decent solution if you want to keep a small set of files synchronized, but doesn't have a whole lot of flexibility for using it in ways for which it wasn't intended.
Microsoft explains this in relation to Vista and the same would apply for its inclusion in Windows 7.