There will always be piracy, just like there will always be shoplifting and theft.
I think the best ways to combat piracy is to make it worthwhile to purchase. Your average person is basically honest and is willing to pay, as long as they believe that they are getting good value for their money.
For music and movies that means things like DRM-free and no restrictions on playback, as well as a reasonable price (for example, iTunes has shown that 99 cents/track seems to be a sweet spot).
A company can also work at developing brand loyalty, which I think might help. Look at companies like Microsoft: We perceive them as big, uncaring, clueless, faceless corporations, reaping multi-millions in sales selling bloated, buggy software. And when you do buy it, you have to go through Activation schemes and other hoops just to get it going. What incentive do I have NOT to pirate a copy, like everyone else does? Or the big media execs, who have made it abundantly clear that they feel the current prices "do not reflect the value of the product" and demand more and more restrictions?
Compare that to Rovio Software, makers of Angry Birds, or the Omni Group (Omni Graffle, Omni Outliner, etc.) or Delicious Software, as a few examples. These people work hard and obviously put a lot of care into the high-quality products they make. Using their software makes me smile. They're not necessarily inexpensive, but there's value for money and so I have no problem paying to support the development work. I might consider the idea of pirating their software, sure, but in the end I want to pay them so I can support their work.
Now it could very well be that the difference is all in my head, but that's still where you want to be. Give your customers a product that they like at a price that they're willing to pay, and don't treat them like thieves, make them feel good about it, and they'll pay. Try to rip them off, make it clear that you're greedy for more, treat them like they're already thieves, and they'll say "well, if you think I'm a thief, I might as well fulfil that expectation".