A western-Pennsylvania bastardization of the phrase "you ones", which addresses more than one person. Since we "western PA people" do things like slur words together, replace almost all vowel sounds with a "schwa" sound (the upsidedown e in the dictionary), replace long "a" sounds with long "e" sounds, and refuse to pronounce any "t" that isn't the first letter of a word, this became "you-ens", then "youns", "yunz", and finally "yinz". Someone must have put this particular pronunciation and spelling in print at some point, since it seems to be the most accepted way to say and write it.
The aforementioned characteristics of our accent are probably at least partly due to the diverse heritage of the population of this area, which is mostly made up of Italian, Irish, Polish, Native American, and African American people, NOT inbreeding, as someone in an earlier entry stated.
The word has become associated with Pittsburgh in particular, whose natives are commonly referred to as "yinzers".
"Are yinz comin' up fer Chrissmiss?"
"What're yinz doin'?"
"I seen yinz at the fleamarket yesterdee."