Do I own a domain if I register it?

No, you do not own the domain name. You are leasing the right to use the domain name for a specific period of time. At the end of that time period you can choose to new the domain or let the domain expire. An expired domain can be registered by another party and used for any purpose they see fit.

Due to the fact that domains are not owned property they can be disputed in certain situations. The ICANN agency regulates all domain registration and usage on the internet. If ICANN receives a claim against a domain name that you have registered then they can transfer the domain registration to another party leaving you without your domain name. Typically this only happens if you have registered a domain name that infringes upon a company’s trademark or registered rights to their intellectual property. For example you shouldn’t register apple.com, should it ever expire and become available, and expect to keep it for any length of time.