I signed a one year lease in New Jersey with a move in date of March 1st. The apartment is $2,000 a month. I paid $2,000 for a security fee (already cashed). I just (January 27) called the landlord (who is the daughter of the owner who is very ill in a home). I told her my grandmother has fallen ill and I will need to move in with my family to help out with Grama. I will be unable to take the apartment at this time. The Landlord said she needed to speak to her father and will get back to me (she's in Mauii until February 3). She did not sound willing to let me off. The apartment is still under construction. Expected to be done by March 1. I have been given a key and was told it was okay to start to move things into the closets and completed rooms before I move in officially in March. I have not started to move things in. I had a delivery planned for this weekend and will cancel that immediately. What is my legal obligation to the lease and will I be able to get my $2000 back?

You have a contract and are bound by it. Your best course of action is to attempt to negotiate a buy out of some sort. You will undoubtedly be responsible until a new tenant is located but the landlord must make every reasonable attempt to find a replacement tenant and mitigate his/her damages.

The law in NJ concerning break a lease due to health of a family member is pretty clear cut: "Death or disability of a tenant or a tenant's spouse
The law recognizes that death or serious illness often requires households to search for less expensive housing, including moving in with other family members. The law provides that any lease for one year or more may be ended before it expires if the tenant or the tenant's spouse dies. The landlord must be given written notice of the lease termination by the tenant or the tenant's executor or administrator, or the surviving spouse if the names of both spouses are on the lease. The lease termination becomes valid 40 days after the landlord receives written notice if (1) the rent owed up to that point has been paid; (2) the property is vacated at least five working days before the 40th day; and (3) the tenant's lease does not prohibit early termination upon the tenant's death. Cite: N.J.S.A. 46:8-9.2. "

LSNJLAW - F: Chapter 5: Ending or Breaking Your Lease (http://www.lsnjlaw.org/english/placeilive/irentmyhome/tenantsrights/chapterfive/index.cfm)