Giving notice of intent to move out is something that should be covered in your lease agreement.  It varies from apartment to apartment, so it is important to read your lease agreement carefully to make sure you give notice to your landlord properly and in a timely manner.  The Austin Tenant’s Council covers the general ways notice must be given.
Either the landlord or the tenant can give notice.  If you don’t want to move, your landlord still has the right to ask the tenant to move out at the end of the lease term.  Neither the tenant nor the landlord need a reason for ending the lease contract.  The exception to this is that a landlord can’t give a tenant a termination notice as punishment for the tenant requesting repairs or as discrimination against the fair housing law.
Most leases stipulate that the lease automatically renews on a month-to-month basis until a 30-day notice is given.  Also, make sure to see if your lease requires written notice rather than verbal notice and plan accordingly.  The lease also may restrict when you can terminate the lease.  They may only allow it to end at the end of a calendar month.  So you will be responsible for the entire month of the lease even if you give notice on the 15th.  

Your written notice of intent to move should have the address of the rental, the date the lease ends, the move-out date planned, and your forwarding address so you can get back your security deposit.  The letter should be given to the landlord or leasing agent in the presence of a witness or by certified mail, return receipt requested.  Be sure to keep a copy of the letter as well.