Registering with the Police
This being China, they want to know where you are at all times. To accomplish this seemingly impossible goal, you are meant to register with the the local police station (PSB) in every new place you go to. You'll also need to re-register if you change accommodation, cities or have your visa extended or get a new one. You won't be able to extend your visa without registering with the police and you will also have to pay a 500 yuan fine if caught without your registration slip 72 hours after arrival.
In Beijing you need to find out where your local PSB is and head down there with some documents. You'll need your passport along with a copy of the visa, your photo page, and the page with your entry stamp on it. They'll keep these.
Living in a Rental Property
You will need to produce your lease to prove where you are living and also a copy of your landlord's ID card. Who knows why they need that. The lease for my place was in my housemate's name but this was no problem.
Living in a Hostel/Hotel
Hostels and hotels seem used to this requirement and will give you a form to fill in.
BEFORE YOU LEAVE HOME
MONEY & COSTS