If you own something, especially something of value, then you have proprietary rights. The word is most often used in relation to new inventions or patents.
Proprietary refers to property: things that are owned by individuals or businesses. People talk about proprietary drugs, proprietary software, and other things that can only be made and sold by those who discovered or created them. A proprietary claim is usually protected by trademark or copyright. When you say you have a proprietary right to something you're saying to everyone else: "Hands off! This is mine." Years ago, owners of small mom-and-pop stores were known as proprietors.