Last month's rent is exactly what it sounds like... the rent payment for the last month on the lease.
HTH

EXAMPLE: Your lease runs from September 1, 2006 to Septembber 1 2007 ... landlord collects "first and last months rent" up front.

That means that you are paying rent for September 2006 and August 2007, plus any deposits. When August rolls around, IF AND ONLY IF you are moving out, you don't pay rent. If you renew, the pre-paid rent rolls over onto the new lease.

Some landlords call the damage/security deposit "last month's rent" and expect you to pay the full rent for the last month (the deposit is of course returnable).