Good kitchen facilities, cleanliness and a nice homely atmosphere. As mentioned in a previous post, the owner/manger also has a lot to do with the quality of a hostel. Probably the best hostel I have ever stayed in was the YHA in Augusta, Western Australia which had all of the above. 
Quoting james
Quoting kezin
At 3: 00 a.m. there is partying in the corridors, in the stairwells, in the dorms themselves, even in the common washrooms. Shrieking, hysterical laughter, running loudly, slamming doors. Naturally, if one of us ( meaning those trying to get some sleep ) tries to ask- in a firm, civil manner- the ' yahoos ' to kindly cease and desist- I or We are told to piss off, or even threatened.
Um, sorry about that
- That's O.K. mate. For my part, sorry for the dislocated shoulder, elbow to the left cheekbone, and knee to the groin you got from me. No hard feelings, I hope ? 
As for me, cleanliness is important.
Privacy too.
Free breakfast (I'm always on a tight budget except when it comes to splurge my money on shopping)
Friendly and informative staff is a bonus!
