Poor print quality and materials, incorrect/missing pieces. I don't know if you've seen counterfeit DVDs before, but they often have badly printed (misaligned, pixelated, even watermarked) graphics and cheap packaging (a basic jewel case with a printed slip of paper vs. a cardboard fold-out, for instance). But at least the DVD itself stands a chance of being a good rip. For a board game, the print quality and materials are kind of what you're paying for.
What djur said, plus some board games you register the serial number on a website. If the serial number is invalid you know you got a counterfeit copy.
