Budget describes the Piccadilly Backpackers very well. If the elevator is out, you'll have to climb the four flights of wind-sucking stairs to get to your tiny room for four.
Our room, while the sheets were clean, was not so nice. It was pretty small for the four of us, and didn't smell all too amazing. We did have our own sink, which was nice, but not our own bathroom. The bathroom down the hall was perpetually leaky and the showers always overflowing. We were situated next to some VERY noisy high-school students from France, who thought that it was fun to yell for no apparent reason.
The location was definitely a huge PLUS. If you want to be central and within walking distance to many areas, shopping, Soho, Leicester Square and other attractions, this could be the right choice for you. There is no lockout or curfew, which is great if you want to check out the clubs.
London is expensive in general, so looking for a bargain can be tough. If you aren't too particular about your sleeping environment, stay at the Backpackers. Just don't forget your earplugs!