I never lived in a B&B before going to York. For a student, it's the best way to save money and experience the British way of living.
I found this B&B from the website available here. It's a great website for you to find accommodations all over the United Kingdom.
Let me tell you why I chose this one from the website, because it looked really nice and cheap. It costs 18-19 pounds per person per night depending on the facilities provided. We enjoyed living here... the rooms are nicely decorated, and we had a great English breakfast.
The host is nice and friendly, he provided us with detailed tourist information. And you can find a map on the wall, so you don't need to worry much about it...
You can get here by taxi from York train station, just costs you 3 pounds... There are plenty of restaurants around here, and you can walk to the city centre within 10 minutes. So, it's a good deal.