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1.  A set of operative facts creating a right enforceable in court.  Though generally synonymous with cause of action, "claim" is slightly broader because it captures not only wrongs with standard names (e.g., battery), but also newly created rights that lack standardized names. 

2.  In patent law, a technical description for each segment of the invention that the patent protects.  Most patents include multiple claims.  The first claim usually describes the whole invention in the broadest terms allowed by the USPTO.  Subsequent claims describe - in increasingly precise terms - how each unit of the invention gets created. 

See Patent infringement.