Sounds like an judgment of eviction and a warrant of eviction were entered against you after a court appearance. If you have a valid defense as to why there should not be a judgment against you, the only thing you can do is hire an attorney to draft an order to show cause to vacate the judgment.
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If you didn't pay the rent, then your landlord can evict you. If you lived there for 6 months, then you're obligated to pay rent for those 6 months.
Based on the facts you've presented, I don't see how an appeal will help you except to buy you more time to find another place to live.
DISCLAIMER: Brandy A. Peeples is licensed to practice law in the State of Maryland. This answer is being provided for informational purposes only and the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship. For legal advice relating to your specific situation, I strongly urge you to consult with an attorney in your area. NO COMMUNICATIONS WITH ME ARE TO BE CONSTRUED AS ARISING FROM AN ATTORNEY-CLIENT RELATIONSHIP AND NO ATTORNEY-CLIENT RELATIONSHIP WILL BE ESTABLISHED WITH ME UNLESS I HAVE EXPRESSLY AGREED TO UNDERTAKE YOUR REPRESENTATION, WHICH INCLUDES THE EXECUTION OF A WRITTEN AGREEMENT OF RETAINER.
If you saying that you need more time in which to move, then you or your attorney can ask for additional time by an order to show cause. You may not get it If you think the court was wrong in entering the judgments of eviction you should consult immediately with an experienced landlord/tenant attorney about seeking a stay from the Appellate Term. If the court was correct, there is little else you can do except to find a new apartment Remember you can only be evicted by a City Marshall and not by your landlord acting on his/her own.
If the judgment for unpaid rent came by an agreement (a stipulation of settlement) you do not have a viable appeal because the judgment is due to your consent. You would need an order from the court vacating the consent judgment and if that is granted, an appeal is no longer an issue.
If the judgment came about because of a default in appearing in court on a scheduled court date, or because the the petition was not answered, there is no appeal. A tenant cannot appeal from a default judgment. The tenant must first make a motion to vacate the default judgment.
If the judgment entered for the unpaid rent is after a trial, you are allowed to seek a stay of the execution and issuance of the warrant of eviction pending an appeal to the Appellate Term 2nd and 11th and 13th Judicial Districts.
Usually, in order to grant a stay of proceeding pending appeal the court will require an "undertaking" and may mean that either the judgment is paid or deposited in court and, going forward to keep the stay alive during the appeal, that the tenant pay use and occupancy to the landlord in the same amount as the rent each month. The stay would could be vacated on default in payment and the eviction may proceed but the appeal could still go forward.
Theoretically the judgment may not be correct. A month to month tenant may have a judgment for possession entered for the last month rent that was not paid as stated in the rent demand and for the subsequent rent for months that accrued during the proceeding, and with oldest unpaid rent treated as a money judgment only, without affecting the tenant's right to redemption (that is not a part of the judgment for possession.) So if that is not what happened, that may be wrong.
I would strongly advise that you consult with an attorney.
Although you may be correct and the judgment was made in an incorrect amount, because you are a month to month tenant who owes rent, there is no real point to an appeal.
If somehow you win the appeal, you would have paid the landlord all the unpaid rent just to avoid eviction during the appeal and to keep possession of the apartment. After all that effort and money spent by you, the landlord may simply end your revived month to month tenancy with a thirty day notice of termination of tenancy.