Leo Courtyard Hostel
This hostel claims to be the cheapest, it has charm, nice people, you'll have to visit the restaurant that it has with really nice employees who will be happy to speak broken English with you and teach you Chinese. The food isn't very Chinese, but it's good, even though all around it there are 100% Chinese restaurants at Chinese prices, like 8 yuanes for some noodles.
This hostel is in the middle of Hutong, which is one of the streets in the old part of town, full of life, shops, restaurants, etc. This hostel is centrally located, walking you can reach Tienanmen, but the real pleasure is losing yourself in Hutong and trying to get to know its people speaking Chinese.