"Piracy" is slang for copyright infringement, the unlawful copying of the work of another, usually for the purpose of distribution and profit. In contrast to swashbuckling pirates of yore, modern day "pirates" make copies, sometimes perfect digital copies, of the work of another. Copyright holders are concerned with infringement because they believe that if someone can buy a copy of their work for less or obtain it for free or through a trade, he or she will, and profits will be lost.
If you are accused of piracy, then someone is claiming that you have unlawfully copied part or all of their work. Computer software, video games, music and DVDs are common objects of copyright infringement actions. If someone can download songs, games or other material that may be copyrighted from your website then you might be accused of piracy. If your site enables users to trade copyrighted material, or if you provide tools that allow others to crack the codes protecting copyrighted material then you might be accused of assisting in piracy, more accurately called "contributory copyright infringement." Regardless of whether the copied materials are offered for sale, distributed for free, or given to relatives, the copyright holder may claim infringement.
Many web sites that have infringing material add a disclaimer stating that the site does not support copyright infringement. While such disclaimers will not necessarily shield you from liability, following the DMCA Safe Harbor may be the best way to protect yourself.