Low quality individual. Basically, any person deemed to be of inferior intellect, common sense, social skills, etiquette, savoir faire, etc. Usage of the term would become appropriate following a demonstrable offense of the above (someone doing something tactless, tasteless, idiotic, or that which breaks either general or local unspoken rules of society).
1. Dave: "What's up with this douche pissing around in the left-hand lane? Move over, gramps!"
Bob: "What an l.q.i."
2. Anne: "I asked you to pick me up something from the liquor store, and you bring me Mad Dog 20/20?! Jesus H. Christ, you're an l.q.i., Phil."