First a quick note: Switzerland in general, and Zurich particularly can be very expensive. It is one of the most expensive countries in whole Europe and because of the strong Swiss franc, the exchange rates aren't very good at the moment. On the other hand, Zurich is quite a small city, and the public transport is quite good and reliable. So you can be in the center quite fast, if you decide to stay a little bit out of Zurich to save some money. Generally, if you want to stay in the city, I would recommend to stay around Langstrasse or Niederdorf if you want to party, and to stay Wipkingen, Altstetten, or Kreis 6 (District 6) if you want to relax a little bit more. Schwamedingen, Affoltern or Seebach are also okay, but they are a little bit further away from the city center.
So if you're looking for bargains, you have essentially two possibilities. Either private accommodations, such as B&B or "cheap" hostels:
Airbnb has also quite a few accommodations for Zurich. The cheapest starts around 10 USD for a night. Basically all neighborhoods inside the city itself are more or less close to the city center, since Zurich is not really big and well-connected by public transport. You might want to avoid the areas Affoltern, Seebach and Schwamendingen.
There are also some hostels in Zurich. If you check hostelworld, private double rooms start around 45 CHF per night if you want to stay in Kloten where the airport is. There are quite a lot of trains to the city center, and it takes around 10-15 minutes. If you really want to stay in the city center itself, you can expect to pa at least 60 CHF. For example this hostel is located in the Niederdorf, where a lot of bars are located and you're in walking distance of all major sights. If you're interested in nightlife and going out, you might also consider to stay in the hostel Langstars. It is located directly over a bar in the Langstrasse where a lot of bars are located. Don't be scared by the prostitutes, drug dealers or drunk people you see there, it is not dangerous at all. But be aware, I would only take an accomodation in this area, if I'm interested in nightlife, because there is always a lot of noise almost 24/7.