Other forms: subletting; sublets
When you rent an apartment by taking over another person's lease, instead of renting directly from a landlord, you sublet. You can also say the person sublets the apartment to you.
As a noun, sublet refers both to the lease agreement and to the property itself: "I'm living in a tiny sublet right next to the train tracks." The formal, legal word for this is sublease, but it's more common for renters to use the informal sublet instead. Being able to sublet to another person makes it possible for a renter to leave an apartment without breaking their lease.