You "own" it, but ownership is not absolute. It's definitely not a lease.
Below is a copy and paste of my reply regarding this same topic on NamePros last year.
http://www.namepros.com/domain-name-...ou-wanted.html
In my view, and that of some U.S. courts, domains are property - hence, you own the domain name...
Think of renewal fees as akin to property tax. Don't pay the renewal, and just like with physical property, one can lose the domain.
Some argue that renewal fees mean that one never truly owns the domain. Well, in a matter of speaking no one, other than governments, own much of anything ... even one's body, in many jurisdictions, is not truly theirs ... ie. laws against committing suicide as well as controls on the sale of body parts, such as kidneys, etc.
One is, in effect, granted a conditional right (ie. paying renewal fees along with other conditions, such as how it's utilized), to a particular domain name by the registry, on behalf of a government entity - registries don't own TLDs, ICANN (which is part of the U.S. Dept of Commerce) "owns" gTLDs while ccTLDs are "owned" by their corresponding governments; ie. TV is owned by the nation of Tuvalu.
Much in the same way that physical property, ultimately, is truly owned by governments. They can, and most often do, dictate how one uses property, such as through zoning, building codes, etc. And can take control of the property anytime, potentially even without offering adequate compensation, if they so choose and are determined enough; eminent domain.
Bottom line, in my view, one basically "owns" a domain name to roughly the same extent one can "own" a piece of land. Ownership is not absolute; better to think of it as the "control" of something for an indeterminate period of time, which is
different than a lease.
Ron