It's awesome. Well, not until you're sharing it with someone. But then it's just great.
Family room lets you share:
The notes part is "just" a One note notebook. We use this for shopping list, both for our weekly run to the grocery store and when we're going to the cabin. Another use case is writing down ideas for presents and packing list. The notebook is also available on Skydrive.
The calendar works as a regular outlook calendar and it is merged in the normal calendar view on the phone. One thing I miss, or perhaps I just have not tried hard enough, is to have my family room calendar added to outlook on my laptop.
When sharing a photo from your phone you can select to share it with familiy room, and it will be available for the other users in high quality.
The chat follows the example from calendar and also integrates smoothly with the phone apps and pops up in the messaging application and also in the notification center.
It makes a great argument for the whole family to run windows phone, or at least those who do not need the latest candy crush and hipster app. Did I say the UX is superb?