Currently you can only play Scratch projects on a web browser (using this site) or using Scratch itself on the desktop. Playing them using this site doesn't fit your needs? If you want to publish them on your own website you can do that as well.
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To play a Scratch file on a website (either this site or your own web page), you don't need to have Scratch installed. Your web browser just needs to support Java. Many browsers already have Java installed, but if yours does not, you can download the Java player for free from:
http://www.java.com/en/download/index.jsp
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