I have to agree with @CyanoticWasp and @johnpowell, @roundsquare and @perspicacious on this one. Unless there are some other issues here it seems a little premature to ask about your legal rights unless maybe this is your first apartment and the first time you’ve ever had a lease expire; but, if you haven’t already discussed renewing your lease with your landlord I wouldn’t start the conversation with a discussion of your legal rights, it’s just not necessary at that point and may make you look like a bit of a troublemaker.
That’s why I asked if you had talked to your landlord. That’s generally the procedure when your lease is about to expire. You contact your landlord and talk to him or her about renewing your lease. Your landlord can choose to let you renew your lease as is, or can change the terms at that point and offer you a new lease under the new terms. The change in terms could involve a rent increase, maybe the duration of the lease (like asking you to sign a 6 month lease or a 2 year lease, e.g.), change in pet policy, etc. Then you either agree to the new terms and sign the lease or not.
At that point if the landlord doesn’t want you there anymore he or she can give you whatever notice is required by the landlord tenant laws in your area and you have to go. If your lease expires without either you or your landlord doing anything about it, as every one has said you transition to a month-to-month tenancy and your rights and your landlords rights are governed by landlord tenant law regarding month-to-month tenancy in your state or area.
Under that kind of tenancy you can continue to live there, as long as you pay your rent, etc, until either you want to move or your landlord wants you to move. You can, upon giving the required notice can move out whenever you want; your landlord, giving the required notice, can ask you to move out anytime he or she wants you to.
And maybe this goes without saying because it’s just common sense (but sometimes I don’t know when to shut up) and I’m not a lawyer, so I don’t know what the laws are regarding whether or not you still legally bound by any of the specific terms of your lease, but whether you are or not, you will still want to comply with them. Say you’re renting a house and your lease expressly prohibited you from having pets and required you to mow the lawn. Going to a month-to-month tenancy doesn’t mean it’s time to go: “Wee! I can finally get a dog and stop mowing the lawn! Yea!” Because that’s going to give your landlord a pretty good reason to want you to leave and you’re probably not going to get your security deposit back when he find he has to replace the dog poop carpet. :-)