Leasehold is a form of property tenure where one party buys the right to occupy land or a building for a given length of time. A lease is a legal estate leasehold estate that can be bought and sold on the open market and differs from a tenancy where a property is let on a periodic basis such as weekly or monthly. At the end of the lease period (often measured in decades - a 99 year lease is quite common) the leaseholder has the right to remain in occupation as an assured tenant paying an agreed rent to the owner. Terms of the agreement are contained in a lease, which has elements of contract and property law intertwined.
Colloquially, a "lease" is often a formalization of a longer, specific period as compared with a "rental" that created a tenancy at will, terminable or renewable at the end of a short period.