Domain ownership is not like a physical object. You never really own a domain.
In exchange for your domain registration fee, you acquire the right to use that domain.
If you fail to pay your fees or do not renew your domain, you release your right to that domain.
ICAAN, who governs domains, has policies that control how this domain is released back into the available domain pool. These are much stricter than they used to be and there's a period of days during which you can re-claim your domain, the registrar can then try to sell it and finally the domain is deleted and becomes generally available again.
Domain name speculators, sometimes known as domainers, often snap up potentially valuable domains.
Domain Leasing (Renting)
You could view your domain registration as a lease but it is in fact very different.
When you register a domain, you become the Registered Name Holder.
This is an important ICANN designation as it is only this name holder who has full rights to the domain.
As a name holder, you can sell the rights to your domain to another party. I recently sold a domain to another party. I transferred my rights to use the domain to them. In return, they paid me.
In contrast, there's an emerging practice of domain leasing.
Under these agreements, you are, in fact, signing a lease. However, you are never the Registered Name Holder. As such, you have no-standing when it comes to domain disputes, you cannot sell the domain rights and the lessor may bail - leaving you without a domain.
So while you could view your domain registration as a lease -- it is important to understand that there's an emerging practice of domain leasing that is very different from domain registration.