It is becoming less and less common, but private rooms in private apartments are definitely an option for budget travelers in Hungary. When you book a private room, you get a room in someone's apartment. You'll usually share the bathroom either with the hosts or with other guests. Breakfast may or may not be included, but the host will often offer a continental spread (bread, butter, jam, coffee or tea) for an extra charge, but it may be cost saving to get your breakfast elsewhere. You may also have limited kitchen privileges (ask in advance to be sure). Some landlords will greet you when you arrive, give you a key, and seemingly disappear; others will want to befriend you, help you change money, show you around, and sometimes provide other services depending on their language skills.
Most rooms are quite adequate; some are even memorable for positive reasons, but any number of reasons may cause you to dislike your accommodations. A location in a noisy or difficult-to-reach neighborhood, a tiny bathroom, or wretched coffee are among the complaints overheard from the occasional displeased traveler. The great majority of guests, though, are satisfied; certainly, staying in a private room provides a window into someone's everyday life in Hungary that you can't really find anywhere else.
You can book rooms through accommodations agencies (for example, Ibusz tel. 1/485-2700; www.ibusz.hu). Prices vary slightly between agencies, but, generally speaking, an average room will cost between 4,500 Ft and 6,500 Ft ($23-$33/£12-£17) for two people, or 5,000 Ft ($25/£13) for a single, plus a 3% tourism tax. High-end rooms in fashionable neighborhoods can cost significantly more. Most agencies add a 30% surcharge (to the first night only) for stays of less than 4 nights. When booking a room, make sure you know its exact location on a map and that you know how to get there. Although there's scarcely an address in Budapest that cannot be reached by some form of public transportation, some are much more difficult to get to than others.