In general, a contract is formed when both sides agree to the same terms. If the landlord offered to rent you the property for $1,000 a month for one year, and you agreed to the monthly charge but wanted a lease for 2 years, there would be no contract because while you agree to the price per month, there was no agreement on the length of the lease. If you signed at the end of a lease in the space provided for your signature, it is generally understood that you agreed to the lease, and you are bound, even if you did not sign each of the pages. If you marked through sections of a lease, and initialed those changes, then signed at the end, the landlord can choose to hold you to your counteroffer, and accept the lease as changed, or reject the lease. It is unclear whether you simply failed to sign certain pages, which could mean you are still bound, or whether you changed the terms such as the pricing, in which case you are not bound, unless the landlord has accepted your changes. You will want to bring a copy of the document to a lawyer for his review.
Note that even if there is no express agreement, there are times that parties are liable on an implied agreement, such as when you accept a good or service understanding it is not offered for free, even though there is no express agreement on the terms. So if you are living in the apartment now, a court could require you to pay for each month that you lived there, at a "reasonable rate" even without a binding lease, though in such a case, you would usually not be liable for rent after you had left the premises.
Answered on Jul 02nd, 2012 at 5:31 PM