What is the correct word to use when you want to rerent an apartment that you are leaving before your lease is up?

To lease the remaining term of your lease agreement is known as a sublease. You would remain on the "hook" for the balance of the term with your existing lease while the sublease tenant would then be obligated to you. You may want to find a replacement tenant and see if the landlord would cancel your lease and then execute a lease directly with the tenant you identified. This would relieve you of any potential liability. Just remember that the landlord is not obligated to do so nor are they obligated to allow your sublease. Check with an attorney to clarify your rights under the existing lease agreement.