Requirements vary based on the owner and the building. The following are a few general guidelines that apply in most cases:

You will need to show an income of at least 40-50 times the monthly rent.
You must show a secure and stable employment history.
You must have good credit history. (For more info and a free credit report go to www.freecreditreports.com)
You must have a social security number. If you do not have a social security number or are not in the process of applying for one you must have a guarantor co-signing your lease.
You must show a good rental history.
All information has to be referenced in form of a letter or sometimes a phone call.
For business owners tax returns will be required.
All students must have a guarantor to co-sign their lease. (See useful terms and definitions.)
Some buildings do not accept students- this is a legal practice and cannot be challenged in court.
If you do not meet the above criteria there are usually two possible solutions:
You can get a co-signer or guarantor (See useful terms and definitions section.) to co-sign the lease for you.
Or some owners will allow you to prepay 6 months to 12 months rent up front.