Relating to the nature or attitudes of a member of a street gang.
This derives from a member of a large set of slang terms invented by Anthony Burgess in his book A Clockwork Orange of 1962. A droog is a young ruffian, or an accomplice or henchman of a gang-leader. The continuing impact of Burgess's novel (and the notorious Stanley Kubrick film made from it) is clear from the way that droog continues to appear in English writing (and has even reached a few dictionaries) and that droogish has been created as a derived term that wasn't in the original vocabulary.
Droog, like much of the slang in the book, is Russian in origin, in this case coming from drug, friend. Burgess called his slang Nadsat, from the ending in Russian of the number words from 11 to 19 (so it's a close equivalent of our teen). Other words from the set that are sometimes seen are malenky, small or little, and poogly, scared. (from World Wide Words)